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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ C ( (^^^ (4L icX err <ir I ^K^HHf. '. c <:<f<rfH rT" ,r<^r'^m ( C6 CS C t i^( Ci 'l^^l : i f^ (I ^^M '" 4 rjcC- '^^^i i. ^ ( 4 r? (■ ( 1 ( C^ vC • I , <#r ..CO m 4: ,-^t r » MATHEMATICS PRACTICAL MEN: BEnre A COMMON-PLACE BOOK or PURE AND MIXED MATHEMATICS, DBSIOVSD OHHILT rOR THB VSE OF CIVIL ENGINEERS, ARCHITECTS, AND SURVEYORS. BY OLINTHUS GBEGOBY, LL.D., F.R.A.S. THIRD EDITION, REVISED AND ENLARGED BY HENRY LAW, cnriL xiranriVB. LONDON: JOHN WEALE, 69, HIGH HOLBORN. 1848. LONDON : OBOROB WOODFALL AND SOX, AJfOCL COURT. 8KINNBR STRBKT. TU SIR ISAMBART BRUNEL, F.RS., na BO. xra THIS WORK. DESIGNED FOB THE USE OF MEMBEBS OF THAT FBOFESSION WHICH HE HAS 80 OBEATLY ADVANCED, IS INSCBIBED, , IN GBATEFUL ACKKOWLEDOMENT OF MANY KINDNESSES, BY HIS OBUOED FBIEND AND PUPIL, HENRY LAW. AUT HOE'S PREFACE. Ths wot\l now presented to the public bad its origin in a desire which I felt to draw up an Essay on the principles and apph'cations of the Tnechanical sciences for the use of the younger members of the InsdtutioQ of Civil Engineers. The eminent individuals who are deserredly r^arded as the main pillars of that useful Institution, stand in need of no such instructions as are in my power to impart : kt it seemed expedient to prepare an Essay, comprised within moderate limits, which might furnish scientific instruction for the many young men of ardour and enterprise who have of late years deToted themseWes to the interesting and important profession, of whose members that InsUtution is principally constituted. My first design was to compose a paper which might be read at one or two of the meetings of that Society ; but, as often happens in such cases, the embryo thought has grown, during meditation, from an essay to a book ; and what was first meant to be a very compendious selection of principles and rules, has, in its execution, assumed the appearance of a systematic aualysis of principles, theorems, rules, and tables. Indeed, tbe circumstances in which the inhabitants of this country are now placed, with regard to the Ioto and acquisition of knowledge impelled me, almost unconscionsly, to such an extension of my ori- ginal plan, as sprung from a desire to contribute to the instruction of that numerous class, the practical mechanics of this country. Besides the early disadvantages under which many of them have laboured, there is another which results from the activity of their pursuits. Unable, therefore, to go through the details of an extensive systematie course, they must, for the most part, be satisfied with imperfect views of theories and principles, and take much upon trust : an evil, how- ever, which tbe establishment of Societies, and the composition of treatises, with an express view to their benefit, will probably soon diminish. Lord Brouorjim, in his ^^ PracHeal Observations upon the Educa^ turn of the Peofk^ remarks that " a most essential service will be ▼1 AUTHORS PREFACE. rendered to the cause of knowle<lge, by him who shall devote his time to the composition of elementary treatises on the Mathematics, sufficiently clear, and yet sufficiently compendious, to exemplify the metliod of reasoning employed in that science, and to impart an accurate knowledge of the most useful fundamental propositions, with their application to practical purposes; and treatises upon Na- tural Philosophy, which may teach the great principles of physics, and their practical application, to readers who have but a general knowledge of Mathematics, or who are even wholly ignorant of the science beyond the common rules of arithmetic/' And again, '^ He who shall prepare a treatise simply and concisely imfolding the doc- trines of Algebra, Geometry, and Mechanics, and adding examples calculated to strike the imagination, of their connection with other branches of knowledge, and with the arts of common life, may fairly claim a large share in that rich harvest of discovery and invention which must be reaped by the thousands of ingenious and active men, thus enabled to bend their faculties towards objects at once useful and sublime." I do not attempt to persuade myself that the present volume will be thought adequately to supply the desiderata to which these pas- sages advert : yet I could not but be gratified, after full two-thirds of it were written, to find that the views which guided me in its execution accorded so far with the judgment of an individual, dis- tinguished, as Lord Brougham was, in early life, for the elegance and profundity of his mathematical researches. With a view to the elementary instruction of those who have not previously studied Mathematics, I have commenced with brief, but, I hope, perspicuous, treatises on Arithmetic and Algebra; a competent acquaintance with both of these being necessary to ensure that ac- curacy in computation which every practical man ought to attain, and that ready comprehension of scientific theorems and formul» which becomes the key to the stores of higher knowledge. As no man sharpens his tool or his weapon, merely that it may be sharp, but that it may be the fitter for use ; so no thoughtful man learns Arithmetic and Algebra for the mere sake of knowing those branches of science, but that he may employ them ; and these being possessed as valuable pre-requisites, the course of an author is thereby facili- tated : for then, while he endeavours to express even common mat- ters so that the learned shall not be disgusted, he may so express the more abstract and difficult that the comparatively ignorant (and the ▲UTHOBS PRBPACI. yil DereliQowledge of Jirithmetie tmdAl^bra is, in our times, compara^ the i^orence,') may practically understand and apply them. Mter the first 97 I>age8, the remaining matter is synoptical. The leoend topics of Oeomctry, Trigonometry, Conic Sections, Curves, Pmpectiye, Mensuration, Statics, Dynamics, Hydrostatics, Hydro- dyiuumcs, and Pneumatics are thus treated. The definitions and pnudples are exhibited in an orderly series ; but investigations and demonstiations are only sparingly introduced. This portion of the voik is aldn in its nature to a syllabus of a Course of Lectures on tiie deportments of science which it treats; with this difference, however, occasioned by tbe leading object of the publication, that popular iUostrations are more frequently introduced, practical appli- eatioDs incessantly borne in mind, and such tables as seemed best odeakied to save tbe labour of Architects, Mechanics, and Civil EngineerB, inserted under their appropriate heads. Of these latter, several have been collected from former treatises, &c., but not a few have been either eompated or contributed expressly for this Common- place Book. In a work like this, it would be absurd to pretend to originality. Tbe plan, arrangement, and execution, are my own ; but the mate- rials have long been regarded, and rightly, as common property. It baa been my aim to reduce them into the smallest possible space, consiatratly with my general object ; but wherever I have found the work in this respect prepared to my hands, I have transcribed it into tbe following pages, with the usual references to the sources from whence it was taken. They who are conversant with the best writers on subjects of mixed nuitbematics and natural philosophy, will know that Smeaion^ Bobison^ Play/air^ Young^ Du Buat^ Leslie^ Hachette^ Blandy Tredgold^ &c., are authors who ought to be consulted, |n the preparation of a volume like this. I hope it will appear that I have duly, yet, at the same time, honourably, availed myself of the advan- tages which they supply. I have also made such selections from my own earlier publications as were obviously suitable to my present purpose ; but not so copiously, I trust, as to diminish the utility of those volumes, or to make me an unfair borrower even from myself. Besides our junior Civil Engineers, and the numerous Practical Mechanics who are anxious to store their minds with scientific facts and principles ; there are others to whom, I flatter myself, the fol- lowing pages will be found useful. Teachers of Mathematics, and those departments of natural philosophy which are introduced into TUl AUTHORS PRBFACB. our more respectable seminaries, may probably find tbis volume to occupy a convenient intermediate station between tbe merely popular exbibitions of tbe trutbs of Mecbanics, Hydrostatics, &c., and tbe larger treatises in wliicb tbe wliole cbain of inquiry and demonstra- tion is carefully presented link by link, and tbe successive portions firmly connected upon irrefragable principles. Wbile students wbo have recently terminated a scientific course, wbetber in our univer- sities, or otber institutions public or private, may, I would fain be- lieve, find in tbis Common -place Book an abridged repository of tbe most valuable principles and tbeorems, and of bints for tbeir applica- tions to practical purposes. Tbe only performances witb wbicb I am acquainted, tbat bear any direct analogy to tbis, are Martin's Young Students Memorial Bookj Jones's Synopsis Pcdmariorum Matkeseos^ and Brunton's Compendium of Mechanics; tbe latter of wbicb I had not seen until tbe present volume was nearly completed. Tbe first and last mentioned of tbese are neat and meritorious productions ; but restricted in tbeir utility by tbe narrow space into wbicb tbey are compressed. Tbe otber, written by the father of tbe late Sir William Jones, is a truly elegant introduction to tbe principles of Mathematics, considering tbe time in wbicb it was written (1706); but as it is altogether theoretical, and is, moreover, now becoming exceedingly scarce, it by no means supersedes tbe necessity^ for such I have been induced to regard it, of a Compendium like tbat wbicb I now offer to the public. In its execution I have aimed at no higher reputation than tbat of being perspicuous, correct, and useful ; and if I shall be so fortunate as to have succeeded in those points, I shall be perfectly satisfied. Olinthus Gregory. Boyal Military Academy, Woolwich, October 1st, 1825. In this new edition I have corrected a few errors wbicb had escaped my notice in the former impression. I have also made a few such additions and improvements as the lapse of time and tbe progress of discovery rendered desirable ; and such as will, I hope, give the work new claims on public approbation. July let, 1883. THE EDITOR'S PREFACE. In presenting to the public a new edition of Dr. Gregory's " Mathe- matics for Practical Men,** the Editor feels that a few introductory remarks are due from him to his readers. Not, indeed, to apologise for presenting to them again a work which they have already marked with their approbation, and which has from time to time been favourably noticed by other writers, but to offer some explana- tion of the numerous alterations which he has taken upon himself to make, and to state his reasons for having done so. In looking through the former edition, it was evident, that since the date of its publication, many of the subjects which it treated upon had been greatly improved and extended ; railways had been brought into successful operation ; the steam engine and machinery generally had undergone vast improvements ; and almost every branch of mechanical science had progressed in some degree ; it was there- fore found necessary, in order to render the present edition as exten- sively useful as its predecessors had been, that the work should undergo an entire revision. In doing this, it was found that so large a quantity of fresh matter would have to be given, as would swell the Tolume to an inconvenient size, and by a necessary increase of price, render the work less accessible to that important class — the working mechanic9*-for whose use it was so peculiarly adapted. To obviate, therefore, these disadvantages, a smaller type has been adopted, so that the work, although not much increased in its exter- nal dimensions, contains more than a third of its present bulk of new matter. In order to add to the usefulness of the work, and render it more generally convenient as a book of reference, a more orderly arrange- ment of its contents has been adopted, and several articles, which, although of much value, did not properly belong to the subject of the work, have been thrown into an Appendix ; in addition to which a Table of Contents has been given, showing at once the general ar- X KDIT0B8 PREPACB. rangement and division of the work, as also a very copious Index at the end. The Editor has endeavoured, in the following short review of the work in its present form, to point out the principal alterations or additions which have heen made : — The work has been divided into two distinct parts ; viz. 1st, Purb Mathematics, comprising chapters upon Arithmetic, Algebra, Geo- metry, Mensuration, Trigonometry, Conic Sections, and the Proper- ties of Curves; and, 2ndly, Mixed or Applied Mathematics, being the application of Mathematics to the general pursuits of the Engineer, the Mechanician, the Surveyor, &c., comprised in chapters upon Mechanics generally. Statics, Dynamics, Hydrostatics, Hydrody- namics, Pneumatics, Mechanical Agents, or Moving Powers, and the Strength of Materials. In the part upon Pure Mathematics : — In the First Chapter, upon Arithmetic, the article npon fractions has been extended, and rules given for the reduction of circulating decimals; and an entirely new section (IX.) is added upon Loga- rithmic Arithmetic, containing a description of the Logarithmic Tables given in the Appendix, with the method of using them, and roles for performing the ordinary processes of calculation by tlieir aid. The Second Chapter, on Algebra, has undergone a very careful re- vision. In Section V., on Involution, a table of roots and powers of monomials has been added ; and in the following section on Evolu- tion, a new rule for finding the roots of powers in general. In Section X. examples have been added of the method for determining the roots of equations. Section XII., on Fractional and Negative Exponents, is entirely new ; as is also nearly the whole of the next section on Logarithms ; to which have been added tables for converting Common Logarithms into Hyperbolic^ and vice versd. The Third Chapter, on Geometry, may almost be said to have been rewritten. The definitions and propositions have been arranged in a more orderly manner, and many additional ones have been added. No demonstration of the several propositions will be found, as the size of the work would not allow, nor did its practical character re- quire, that such should be given; but for the convenience of those who may wish to see them demonstrated, a reference has been made to BDIT0B8 PEBFACB. Xl Euclid, giving the number of the theorem and book in which the demonstration of the same proposition will be there found. The Fourth Chapter, on Mensuration, has also been almost entireiy rewritten : the tables of weights and measures have been put under a more conYenient form, and rules and formuIfiB giveft for the mensura- tion of all kinds of superficies and solids. In the Fifth Chapter, on Trigonometry, yery exteneiTe 2Vt^ofio- metrieal Formulm have \eea added, which cannot fail to be found of serrioe in the pursuits of Engineers and Surveyors. The second sec- tion is entirely new, being a description of the Tables of Logarithmic sines, cosines, tangents, cotangents, secants, and cosecants given in the Appendix, with the method of using them. The Sixth Chapter, on Conic Sections, has been almost entirely rewritten, and several additional formuls and problems relating to them have been added. The Seventh Chapter, on the Properties of Curves, has also been much extended ; the first section of definitions, and the latter part of the fourth section, on the Epicycloid, being entirely new. In the second division of the work, upon Mixsd Mathbm atigs:—- In the First Chapter, on Mechanics in General, many fresh defi- nitions have been added. In the Second Chapter, the whole of the definitions and formulfs in the first section, on Statical Equilibrium, are new. In the second section, the principle of parallel presmret is applied to determine the position of the center of gravity, and an example is given of the centro-baryc method of determining the superficies or solidity of surfaces or solids. In the third section, on the application of Statics to the equilibrium of structures, the formuln have been more conve- niently arranged, and tables have been added of the natural slope assumed by various kinds of materials, as also of the limiting angle and coefficient of friction of the various materials used in the con- struction of arches, &c, the latter extracted from Professor Moseley's work on the Mechanical Principles of Engineering. An article is also added upon Suspension Bridges, a subject of much importance to Engineers, from the general use into which they are being brought. Formultt are given for determining all their elements and propor- tions, the use of which is illustrated by an example. In the Third Chapter, on Dynamics, the definitions in the first section have been nuidi ezlended ; and at the commencement of the second section, on XU EDITORS PREFACE. the General Laws of Uniform and Variable Motion, it has been en- deayoured to put in a clear light the long disputed question relative to the momentum and vis viva of moving bodies. In this section, also, formulse have been added expressing the relation between the times, velocities, spaces, &c. of bodies in motion. In the third sec- tion some practical formulce are added for determining the proportions for the coniccd governor^ so generally used for regulating the velocity of steam engines and water wheels. In the fifth section, on the Mechanical Powers, their true nature is explained, in order to pre- yent any misconception which might be occasioned by their designa- tion ; and a table is added exhibiting the ratio in each, between the power exerted and the effect produced. In the Fourth Chapter, on Hydrostatics, the fifth section, on Capil- lary Attraction, is entirely new. In the Fifth Chapter, on Hydrodynamics, several additional for- malse are given, in the second section, for determining the velocity and discharge through conduit pipes ; and the whole of the formulse have been arranged in a more orderly manner, and numbered for the convenience of reference. In the Sixth Chapter, on Pneumatics, at the conclusion of the first section, a very exact formul«B of Mr. Galbraith's is given, for determining the velocity of sound, and for measuring distances by the same. In the Seventh Chapter, on the Mechanical Agents, the third sec- tion, on the Steam Engine, is entirely new. The nature and pro- perties of steam are explained, and formulse and rules are given, and illustrated by examples, for determining its pressure or temperature. The subject of the specific heat of steam is also explained, and for- mulse given for determiHing it from the pressure; and a table is added of the pressure, temperature, and specific volume of steam, from 5 lbs. to 1 50 lbs. upon the square inch. A general description of the mode of action of the steam engine is given, which is fol- lowed by an investigation of its general theory upon the principles first laid down by the Comte de Pambour. The various kinds of engines employed are then classified, and described separately ; the principles previously educed are applied to each ; and formulse are deduced for the velocity of the engine, the quantity of steam used, and the useful effect of the engine, for each particular case, which are given both in a general and a more practical form. In this por- tion of the work, in treating of engines acting expansively, it was EDITORS PREFACE. Xli) found imposnble to dispense with the use of the Differential Calculus, the introduction of which had heen studiously avoided, in order that the work might be accesable to those who had not entered upon the higher branches of Mathematics. The use of the Differential Cal- culus, howeyer, in the present case, while it was necessary for the proper investigation of the subject, will not, it is hoped, in any way abridge the usefulness of the work to those who are not conversant wiUi its principles, since the results obtained are expressed in the ordinary form, while for those who do understand them, it was thought that it would prove much more satisfJEictory to them to be enabled to examine and satisfy themselves of the truthfulness of the several investigations. The Eighth Chapter, on the Strength of Materials, has been almost entirely rewritten. This portion of the work has been rendered much more valuable by the introduction of the results of Professor Hodgkinson and Mr. Cubitt's researches upon this subject, «o important to the Civil Engineer and Architect. The woodcuts have been replaced by copperplate engravings, and several additional plates have been given, for three of which the Editor is indebted to Mr. R. A. Rumble. An Appendix has been added, containing a Table of Logarithmic Differences, in a new, and, it is hoped, more convenient form than that usually adopted, and Tables of the Logarithms of Numbers, and of Logarithmic sines, tangents, &c. : these tables will be found of great utility to Engineers and others, much engaged in calculations. It also contains a new Table ( VL) of various useful numbers, with their logarithms, and a Table (XI.) of the weight of materials fre- ijuently employed in construction. The remaining portion of the Appendix consists of matter which stood in the body of the former edition of the work, but which has now been put into the Appendix, because, although too valuable to be omitted, it did not properly be- long to, but was only incidentally connected with, the subjects therein treated. The Editor has thought it t>nly doing justice to the late Dr. Ore- gory to point out thus in detail the extent of the alterations which he has made, in order that the public may know how far each is re- sponsible for the work in its present form, and to prevent either blame being attached to Dr. Gregory, or credit given to the Editor, where both might have been misplaced. H. L. London, 21ft October, 1847. CONTENTS. PART I.— PUKE MATHEMATICS. CHAPTER I. Arithmetic. PACK Sect. 1. Definitions and Notation. 1 2. Addition of Whole Numben 5 S. Subtraction of Whole Numben 6 4. Multiplication of Whole Numben 7 5. DiTinon of Whole Numben 10 Proof of the fint Four Rules of Arithmetic . . . .Id 6. Vulgar Fractions 18 Reduction of Vulgar Fractions . . . . . .14 Addition and Subtraction of Vulgar FVactions . .17 Multiplication and Division of Vulgar Fractions . . 17 7. Decimal Fractions 18 Reduction of Decimals 19 Addition and Subtraction of Decimals 22 Multiplication and Division of Decimals 22 8. Complex Fractions used in the Arts and Commerce . .23 Reduction 28 Addition 24 Subtraction and Multiplication 25 Division * . .26 Duodecimals 27 9. Powen and Roots 28 Evolution 29 10. Ph)portion 82 Rule of Three 84 Determination of Ratios 87 11. Logarithmic Arithmetic ^ Use of die Tablet 42 Multiplication and Division bv Logarithms . . • . ^ Proportion, or the Rule of l^ree, by Logarithms . . 46 Evolution and Involution by Lo^urithms .47 12. Properties of Numben 49 CHAPTER IL Algbbba. 8acr. I. Definitions and NoCatioti A2 2. AdfidoD and Subtraction ^ XVI CONTENTS. PAGE Sect. S. Multiplication 56 4. Division 58 5. Involution 60 6. Evolution 63 7. Surds 65 Reduction 65 Addition, Subtraction, and Multiplication .... 68 Division, Involution, and Evolution 69 8. Simple Equations 70 Extermination 73 Solution of Genera] Problems 75 9. Quadratic Equations 77 10. Equations in General 81 11. Progression 84 Arithmetical Progression 85 Geometrical Progression 86 12. Fractional and Negative Exponents 88 13. Logarithms 90 14. Computation of Formulae 95 CHAPTER in. Geometry. Sect. 1. Definitions 98 2. Of Angles, and Right Lines^ and their Rectangles . . .99 a Of Triangles 100 4. Of Quadrilaterals and Polygons 102 5 Of the Circle, and Inscribed and Circumscribed Figures . .104 6. Of Planes and Solids 109 7. Practical Geometry 112 CHAPTER IV. Men&uration. Sect. 1. Weights and Measures 119 1. Measures of Length 120 2. Measures of Sur&e 12] 3. Measures of Solidity and Capacity 121 4. Measures of Weight 123 5. Angular Measure 124 6. Measure of Time 124 Comparison of English and French Weights and Measures . 125 2. Mensuration of Supeirficies ] 27 3. Mensuration of Solids 130 CHAPTER V, Trigonometry. Sect. 1. Definitions and Triffonometrical Formulae 134 2. Trigonometrical Tables 139 CONTENTS. XVU PAGE SxcT. Sw Geoenl Propoeitkms 146 4. Solution of the Casei of Plane Triangles 148 Rifffat-angled Plane Triangles 151 5. On Uie application of Trigonometry to Measuring Heights and Distances 152 Detenninatipn of Heights and Distances by Approximate Me- chanical Methods 156 CHAPTER VL CoNie Sections. 8xcT. 1. Definitions 162 2. Properties of the Ellipse 165 Problems relating to the Ellipse 167 3. Properties of the Hyperiwla 168 Problems relatine to Hyperbolas 170 4. Properties of the Parabola 171 Problems relating to the Parabola 173 CHAPTER VIL PROFEBTISS OF CUETES. Sect. 1. Definitions 175 2. The Conchoid 176 a The Cissoid 177 4. The Cycloid, and Epicycloid 178 6. The Quadratriz 179 & The Catenaiy 180 tenaiy . I of Relat Tables of RelatioDs of Catenarian Curves .... 185 PAET n.— MIXED MATHEMATICS. CHAPTER I. Mechanics in Oxneeal 187 CHAPTER II. Statu». Sect. 1. Statical Eouilibrium 180 2. Center of Gravi^ 102 3l General application of the Prmciples of Statics to the Equili- brium 01 Structures 196 Equilibrium of Piers or Abutments 196 Pressure of earth against Walls 198 Thickness of Walls 201 Bquilibrinm of Polygons 202 Stability of Arches .' . s . .205 Bqpdilicium of Suspension Bridges 207 h XVUl CONTENTS. CHAPTER III. Dynamics. PAGE Sect. 1. Oenenl Definitions 211 2. On the General Laws of Uniform and Variable Motion 212 Motion uniformly Accelerated 214 Motion of Bodies under the Action of Gravity . .214 Motion over a fixed Pulley 216 Motion on Inclined Planes 217 3. Motions about a fixed Center, or Axis 220 Centers of Oscillation and Percussion 220 Simple and Compound Pendulums 221 Center of Gyration, and the Principles of Rotation . 229 Central Foroes 232 Inquiries connected with Rotation and Central Forces . 234 4. Percussion or Collision of Bodies in Motion .... 236 5. On the Mechanical Powers 239 Levers 240 Wheel and Axle 242 Pulley 243 Inclined Plane 244 Wedge and Screw 243 CHAPTER IV. Hydrostatics. Sect. 1. General Definitions 248 2. Pressure and Equilibrium of Non-elastic Fluids .... 249 3. Floating Bodies 254 4. Specific Gravities 256 5. On Capillary Attraction 258 CHAPTER V. Hydrodynamics. Sect. 1. Motion and Effluence of Liquids 260 2. Motion of Water in Conduit Pipes and Open Canals, over Weirs, «6C 262 Velocities of Rivers 265 3. Contrivances to Measure the Velocity of Running Waters . . 273 CHAPTER VI. Pneumatics. Sect. 1. Weight and Equilibrium of .Air and Elastic Fluids . . . 278 2. Machines fur Raising Water by the Pressure of the Atmosph ere 281 3. Force of the Wind 292 CONTENtS. Xix CHAPTER VIL BIechanical Agents. PAGE Sect. 1. Water as a Mechanical Af^nt 294 2. Air as a Mechanical Agent 297 Coulomb's Experiments 801 dw Mechanical Agents depending upon Heat The Steam Engine . 802 Table of Pressure and Temperature of Steam . . .811 General Description of the Mode of Action of the Steam Engine 812 Theory of the Steam Engine 816 Description of the various kinds of Engines, and the Formulffi for calculating their Power 822 Practical application of the foregoing Forroulie . 848 4. Animal Strength as a Mechanical Agent 848 CHAPTER VIIL Strength of Materials. SaT. 1. Results of Experiments, and Principles upon which they should be practically applied 870 2. Strength of Materials to Resist Tensile and Crushing Strains 874 Strength of Columns 879 3. Elasticity and Elongation of Bodies subjected to a Crushing or Tensile Strain 881 4. On the Strength of Materials subjected to a Transverse Strain 885 Longitudinal form of Beam of uniform Strength . 886 Transverse Strength of other Materials than Cast Iron . 887 The Strength of Beams according to the manner in which the Load is distributed 888 a. Elasticity of Bodies subjected to a Transverse Strain . . 888 6. Strength of Materials to resist Torsion 892 APPENDIX. L Table of Logarithmic differences 1 n. Table of Logarithms of Numbers, from 1 to 100 .... 1 HL Table of Logarithms of Numbers, from 100 to 10,000 ... 1 IV. Table of Logarithmic Sines, Tangents, Secants, &c. .19 V. Table of Usoiil Factors, extending to several places of Decimals . 64 VL Table of various Useftil Numbers, with their Logarithms ... 66 VIL A Table of the Diameters, Areas, and Circumferences of Circles and also the Sides of Equal Squares 66 VIII. Table of the Relations of the Arc, Abscissa, Ordinate and Subnormal, in the Catenary 78 IX. Tables of the Lengths and Vibrations of Pendulums ... 76 X. Table of Specific Gravities 80 XL Table of Weight of Materials frequently employed in Construction . 82 XIL Principles of Chronometers 82 XIII. Select Mechanical Expedients 89 XIV. Obaervarions on the Effect of Old London Bridge on the Tides, &c. 92 XV. Professor Parish on Isometrical Perspective -Wi LIST OF PLATES. Platb L' IL IIL V General Diafframs. IV. ^ V. VI. Detiili of a Breast Water-wheel. VI L Fenton, Murray, and Wood*a Steam Engine. VII L Higb-prenure Expansive Engine, by Middleton. IX. Details of Steam Engines. X. Longitudinal Section of a Locomotive Engine. XL Two Transferee Sections of the Same. XIL Details of various Engines. XIIL Isometrical Perspective. COMMON-PLACE BOOK, ETC., ETC. PART I. PURE MATHEMATICS. CHAP. I. ARITHMETIC. Sect. I. Definitions and Notation. Arithmetic is the science of numbers. We give the name of number to any assemblage of units^ or of any parts of an assumed unit; a unit being the quantity which, among all those of the same kind, forms a whole which may be regarded as the hose or element. ' Thus, when we speak of one kouse^ one guinea^ ^ we speak of units^ of which the first is the thing called a house, the fiecond that called a guinea. But when we w,y four houses^ ten gmnea^^ three quarterM ofa-guinea^ we speak of numbers^ of which the first is the unit house repeated four times; the second is the unit QHinea repeated ten times; the third is the fourth part of the unit guinea ^peated three times. In cv«y particular classification of numbers, the unit is a measure taken aroitrarily, or established by usage and convention. Numbers formed by the repetition of an unbroken unit are called vhole numbers^ or integers^ as seven miles, thirty shillings: those which are formed by the assemblage of any parts of a unit are called fractional numbers^ or %\mi^\yf radians; as two thirds of a yard, three eighths of a mile. When the unit is restricted to a certain thing in particular, as one man^ one horse^ one pounds the collection of many of those units is called a concrete number, aa ten men^ twenty horses^ fifty pounds. But if the unit does not denote any particular thing, and is expressed ^mply by one^ numbers which are constituted of such units are deno- minated dieerete or ahstracty as five^ ten, thirty. Hence, it is eyident that abstract nmnbers can only be compared with their unit, as con- B ARITHMETIC : NOTATION. [part I. Crete numbers are compared with, or measured by, theirs; but that it is not possible to compare an abstract with a concrete number, or a concrete number of one kind with a concrete number of another ; for there can exist no measurable relations but between quantities of the same kind. The series of numbers is indefinite; but only the first nine of them are expressed by different characters, called figures; which are as follows : — Names, one, two, three, four, ^sc^ six, seven, eight, nine. Figures. 1, 2, 3, 4, 5, C, 7, 8, 9. These are called sig?iificant figures, in contradistinction to another character employed, namely 0, called the cypher or zero; which has no particular value of itself, but by its positmi is made to change the value of any significant figures with which it is connected. In the system of numeration now generally adopted, and which is borrowed from the Indians*, an infinitude of words and characters is * As the Roman notation is not unfrequently met with, especially in dates, we subjoin the following brief account of it: — The Romans employed only seven numerals, being the following capital letters of their alphabet ;— viz. I, for one ; V, for five; X, for ten; L, for fifty ; C, for an hundred ; D, for five hundred; M, for k thousand ; and for expressing any intermediate or greater numbers they employed various repetitions and combinations, the principles of which are shown in the following examples : — As often as any character is repeated, so many times is its value repeated. A less character before a greater diminishes \ its value. ( A less character after a greater increases ( its value. I ir III 4 = nil or IV 5 6 < 7 •- 8 9 10 40 50 m 100 500 1000 2000 5000 0000 10,000 50,000 00,000 100,000 1,000,000 2,000,000 &c. V VI VII VIII IX X XL L LX C Dor lo M or Clo MM V or loD yi X or CCI33 W I J33 LX CorCCCl033 M or CCCCI3303 MM &c. 5 For every f) annexed, this becomes ten ( times as many. \ For every C and ^), placed one at each ( end, it becomes ten times as much. S A bar over any num1)er increases its value \ 1000-fold. CHIP. I.] arithmetic: notation. 3 avoided, by a simple yet most ingenious expedient, which is this : — (urjifyHre placed to t/te left of another assumes ten times the wUue (to \i rtoM have if it occupied the place of the latter. Thus, to express the number that is the sum of 9 and 1, or ten units, (called ten^) we place a 1 to the left of a 0, thus 10. So again the sum of 10 and 1, or eleven^ is represented by 11 ; the sum of 11 ud 1, or of 10 and 2, (called twelWy) is represented by 12 ; and so on for tiwieen, fourteen^ fifteen^ &c., denoted respectively by 13, 14, 15, &c, the figure 1 being all along equivalent to ten^ because it oecapies the second rank. la like manner, twenty^ twenty^one^ twenty-two^ &c., are repre- sented by 20, 21, 22, because the 2 in the second rank is equivsJent to twice ten, or tteenty. And thus we may proceed with respect to the numbers that fall between twenty and three tens or ^irty (30), four tens or forty (40), five tens or Jifily (50), six tens or sixty (60), RTen tens or seventy (70), eight tens or eiyhty (80), nine tens or iit)ie^(90). After 9 are added to the 90 (ninety), numbers can no longer be expressed by two figures, but require a third rank on the left hand of tbe second. Tbe figure that occupies the third rank, or of hundredths^ is ex- pressed by the word hundred. Thus 369, is read three hundred and sixtj-nine ; 428, Is read four hundred and twenty-eight ; 837, eight kundred and thirty-seven : and so on for all numbers that can be re- presented by three figures. Bat if the number be so large that more than three figures are required to express it, then it is customary to divide it into periods of tkree figures each, reckoning from the right hand towards the \eh, and to distinguish each by a peculiar name. The second period is called tbat of thousands^ the third that of millions^ the fourth that oi milliards or billions*^ the fifth that of trillions^ and so on ; the terms units, tens, ud hundreds, being successively applied to the first, second, and third ranks of figures from the right towards the left, in each of these periods. Thus, 1,111, is read one thousand one hundred and eleven. 23,456, twenty-three thousands, four hundred and fifty-six. 421,835, four hundred and twenty-one thousands, eight hundred •nd thirty-five. 732,846,915, seven hundred and thirty-two millions, eight hundred tnd forty-six thousands, nine hundred and fifteen. The manner of estimating and expressing numbers which we have here described is conformable to what is denominated the decimal notation. But, besides this, there are other kinds invented by philo- sophers, and others indeed in common use : as the duodecimal^ in ' It has been cascmnary in England to give the name of LiHums to millions of millions, of triUknu to millions of millions of millions, and so on : but the method here given of dividing numbers into periods of three figures instead of six, it universal (m the Continent ; and, as it seems more simple and uniform tian the other, I have adopted it. B 2 4 arithmetic: notation. [part i. which every superior name contains (tcelve units of its next inferior name ; and the sexagesimal^ in which sixty of an inferior name are equivalent to one of its next superior. The former of these is em- {)loyed in the measurement and computation of artificers' work ; the atter in the division of the circle, and of an hour in time. To the head of notation we may also refer the explanation of the principal symbols or characters employed to express operations or results in computation. Thus, The sign + (plus) belongs to addition, and indicates that the numbers between which it is placed are to be added together. Tlius, 5 + 7 expresses the sum of 5 and 7, or that 5 and 7 are to be added together. The sign — (mintis) indicates that the number which is placed after it is to be subtracted from that which precedes it. So, 9—3 denotes that 3 is to be taken from 9. The sign ^^ denotes diffei-ence^ and is placed between two quan- tities when it is not immediately evident which of them is the greater. The sign x {into\ for multiplication, indicates the product of two numbers between which it is placed. Thus 8x5 denotes 8 times 5, or 40. The sign -f- {hif)^ for division, indicates that the number which precedes it is to be divided by that which follows it ; and the quotient that results from this operation is often represented by placuig the first number over the second with a small bar between them. Thus, 15 -r- 8 denotes that 15 is to be divided by 8, and the quotient is ex- pressed thus y . The sign =, two equal and parallel lines placed horizontally, is that of equality. Thus, 2 -|- 3 + 4 = 9, means that the sum of 2, 3, and 4, is equal to 9. Inequality is represented by two lines so drawn as to form an angle, and placed between two numbers, so that the angular point turns to- wards the least. Thus, 7^4, and A > B, indicate that 7 1^ greater than 4, and the quantity represented by A greater than the quantity represented by B ; and, on the other hand, 3 < 5 and C < D indicate that 3 is less than 5, and C less than D. Colons and double colons are placed between quantities to denote their proportionality. So, 3 : 5 : : 9 : 15, signifies that 3 are to 5 as 9 are to 15, or | zi yj. The extraction of roots is indicated by the sign v/, with a figure occasionally placed over it to express the degree of the root ; or by a fraction (having unity for its numerator, and the figure expressing the degree of the root for its denominator) placed above and to the right of the quantity to have its root extracted ; thus, v/ 4 or 4*, signifies the square root of 4 ; V27, or 27*, the cube root of 27; V 16 or 16*, the fourth or biquadrate root of 1 6. The raising of powers is expressed by a whole number similarly placed, the figure denoting the, power to which the quantity is to be raised ; thus 6'. signifies the square of 6 ; 8', the cube of 8 ,* and 3\ CHAP. I.] ADDITION OF WHOLE NUMBERS. 5 the fourth power of 3. The figures thus used to indicate the power, whether whole or/ractional numbers, are termed indices or exponents. When both operations are to be successively performed upon a quantity, that is, when some root is to be extracted, and then that root to be raised to some different power, the operation is very simply ex- pressed by a fraction placed as before, the denominator of which indi - cates the root to be extracted, and the numerator the power to which that root is to be afterwards raised. Thus 27*, denotes that the cube root of 27 is to be extracted, and that the root so obtained is to be squared, or raised to the second power. It is immaterial which operation is first performed, for the result would be the same whether we first extracted the cube root, as above, and then squared it^ or whether we squared the number first, and then extracted the cube root of the power so obtained. Although the above signs * are principally employed in Algebra and the higher branches of Mathematics, they are given here, as their use in Arithmetic frequently affords brevity without a sacrifice of perspicuity. Sect. II. Addition of Whcle Numbers, Addition is the rule by which two or more numbers are collected into one aggregate or sum. Suppose it were required to find the sum of the numbcre 3731, 349, 12487, and 54. It is evident that if we computed separately . the sums of the units, of the tens, of the hundreds, of the thousands, &c^ their combined results would still amount to the same. We should thus have 15 thousands + 14 hundreds + 20 tens + 21 units, or 15000 + 1400 + 200 + 21 ; operating again upon these, in like manner, rank by rank, we should have 1 thousands + 6 thou- sands + 6 hundreds + 2 tens + 1, or 16621, which is the sum required. But the calculation is more commodiously effected in the following manner : — Rtde. — Place the given numbers under each other, so that units stand under units, tens under tens, hundreds under hundreds, &c. Add up all the figures in the column of units, and observe for every ten in its amount to carry one to the place of tens in the second column, putting the overplus figure in the first column. Proceed in the same manner with the second column, then with the third, and so on till alJ the columns be added up : the figures thus obtained in the several amounts indicate, according to the rules of notation, the sum required. Nate. — Whether the addition be conducted upwards or downwards, the result will be the same ; but the operation is most frequently con- ducted by adding upwards. * There are other signs employed in the processes of Algebra, an explanation of which will Ue found at page 52, st icq. 6 SUBTBACTION OF WHOLE NUMBBBS. [PABT I. Example. — Taking the same numbers as before, and 3731 disposing them as the rule directs, we have 4 + 7 + 9 + 349 1 =z 21, of which we put down the 1 in the place of uniU^ 12487 and carry the 2 to the tens : then 2-h5 + 8 + 4-f3= 54 22, of which we put down the left hand 2 in the place of tens^ and carry the other to the hundreds : then 2 + 4 + 1662 1 3 + 7 = 16, of which the 6 is put in the hundreds^ and =*^- the 1 carried to the thousands. This progress continued will give the same sum as before. Other Examples, 57 762 5389 97615 III. 6475 9830 2764 5937 77756 3388 9763 90257 10376786 789632 1589 73 103823 25006 181164 11168080 Sect. Subtraction Oj f mole Numbers. Subtraction is the rule by which one number is taken from another, so as to show the difference, or excess. The number to be subtracted or taken away is called the subtrahend ; the number from which it is to be taken, the minuend; and the quan- tity resulting, the remainder. Rule. — Write down the minuend, and beneath it the subtrahend, units under units, tens under tens, and so on. Then beginning at the place of units, take each figure in the subtrahend from its correspond- ing figure in the minuend, and write the difference under those figures in the same rank or place. But if the figure in the subtrahend be greater than its corresponding figure in the minuend, add ten to the latter, and then take the figure in the subtrahend from the sum, putting down tlie remainder, as be- fore ; and in this case add 1 to the next figure to the le/t in the sub- trahend, to compensate for the ten borrowed in the preceding place. Thus proceed till all the figures are subtracted. Example: . . Minuend 26565874 Subtrahend 9853642 Remainder 16712232 Here the five figures on the right of the subtrahend are each less than the corresponding figures in the minuend, and may therefore be taken from them, one by one. But the sixth figure, viz. 8, cannot be taken from the 5 above it. Yet, as a unit in the seventh place is equivalent to 10 in the sixfhy this unit borrowed (for such is the CHAP, I.] MULTIPLICATION OP WHOLB NU1IBBB8. 7 technical word here employed) makes the 5 become 15. Then 8 taken from 15 leaves 7, which is put down ; and 1 is added to the 9 in the seventh place of the subtrahend, to compensate or balance the 1 which was borrowed from the seventh pkwjc in the minuend. Re- course must be had to a like process whenever a figure in the sub- trahend exceeds the corresponding one in the minuend. Other Examples, From 8217 From 44444 Take 21498 Take 45624 Take 3456 Take 3456 From 76262 From 80200 Remains 4761 Remains 40988 Remains 54764 Remains 34576 iVbte. — Although it is customary to place the minuend above the subtrahend, this is not absolutely necessary. Indeed, it is often con- venient in computation to find the difference between a number and a greater that naturally stands beneath it : it is, therefore, expedient to practise the operation in both ways, so that it may, liowever it occurs, be performed without hesitation. Sbct. IV. MuUiplication of Whole Numbers. Multiplication of whole numbers is a rule by which we find what a given number will amount to when it is repeated as many times as are represented by another number *. The number to be multiphed, or repeated, is called the mtUiipli- candj and may be either an abstract or a concrete number. The number to be multiplied by is called the multiplier^ and mtist be an abstract number, because it simply denotes the number of times the multiplicand is to be repeated. Both multiplicand and multiplier are called yac^or«. The number that results from the multiplication is called the pro- duct. Before any operation can be performed in multiplication, the learner must commit to memory the following table of products, from 2 times 2, to 12 times 12. * This definition, though not the most scientific that might be given, is placed Ymbt^ because others depend, implicitly if not explicitly, on proportion, and Iharafore cannot logically be introduced thus early in the course. MULTIPLICATION OF WHOLE NUMBERS. [part I. times 2 3 4 5 6 7 8 9 10 11 12 24 2 4 6 8 10 12 14 16 18 20 22 3 4 6 9 12 15 18 21 24 27 30 33 36 8 12 16 20 1 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 88 84 96 8 16 24 32 40 48 56 64 72 80 9 18 27 36 40 45 54 63 72 81 90 99 108 10 20 30 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 144 12 24 3G 48 60 72 84 96 108 120 132 It is very advantageous, in practice, to have this table carried on, at least intellectually, to 20 times 20. All the products to this extent are easily remembered. The learner will perceive that in this table 7 times 5 is equal to 5 times 7, or 7 X 5 =: 35 = 5 X 7. In like manner that 8 X 3 = 24 = 3 X 8, 4 X 11 = 44 z= 11 X 4, and so of other products. This is often made a subject of formal proof, as well as that 3x5x8 — 3x8x5 = 5x3x8 = 5x8x3, &c. But to attempt the demonstration of things so nearly axiomatical as these is quite unnecessary. Previously to exhibiting the rules for performing multiplication, let us take a simple example, and multiply 4827 by 8. Here placing the numbers as in the margin, and multiplying in their order 7 imits by 8, 2 tens by 8, 8 hundreds by 8, 4 thousands by 8, the several products are 56 units, 16 tens, 6.4 hundreds, 32 thousands : these placed in their several ranks, according to the rules of notation, and then added up, give for the sum of the whole, or for the product of 4827 multiplied by 8, the number 38616. 4827 8 56 16 64 32 38616 CHAP. I.] MULTIPLICATION OP WHOLE NUMBERS. 9 Or the same example may be worked thus : — 8 X 7 = 56 \ Sx 800= eJoo/ ^^^"^^^«^I^^^°^^3^i*^^ 8 X 4000 = 32000 i f™® '" ^"^^^ ^ ^''' I fore. 38G16 /' Cask I. — To multipltf a number^ consisting of several figures^ by a Mumher not exceeding 1 2. Ruk. — Multiply each figure of the multiplicand by the multiplier, beginning at tlie units ; write under each figure the units of the pro- duct, and carry on the tens to be added as units to the prodqct fol- lowing. Examples, Mnltiplv 4827 218043 440052 8765400 Bv ' 8 9 11 12 Products 38616 1962387 4840572 105184800 Case II. — To perform multiplication token each /actor exceeds 1 2. Rule. — Place the factors under each other (usually the smallest at bottom), and so that units stand under units, tens under tens, and so on. Multiply the multiplicand by the figure which stands in the unit's place of the multiplier, and dispose the product so that its unit's place ^hftll stand under the unit of the multiplicand ; then multiply succes- sixely by the figure in the place of tens, hundreds, &c., of the multi- plier, and place the first figure of each product under that figure of the moltiplicr which gave the said product. The sum of these products will be the product required. Example, Multiply 8214356 by 132. Multiplicand 8214356 Multiplier 132 8214356 X 2 = 16428712 8214356 X 3 tens = 24643068 8214356 X 1 hundred = 8214356 8214356 X 132 = 1084294992 10 DIVISION OF WHOLB NUMBBB8. [PART I. Other Example$. Multiply 821436 Multiply 8210075 by 672576 by 420306 4928616 49260450 5750052 24630225 4107180 16420150 1642872 32840300 5750052 4928616 Product 3450743782950 Product 552478139136 Note, — Multiplication may frequently be shortened by separating the multiplier into its component parts or factors, and multiplying by tbem in succession. Thus, since 132 times any number are equal to 12 times 11 times that number, the first example may be performed in this manner : Multiply 8214356 by 11 Here one line of multi- And this product 90357916 ,» plication, and one of by 12 i addition, are saved. Product as before 1084294992 / So, again, the multiplier of the second example, viz. 672576, divides into three numbers, 600000, 72000, and 576; where, omitting the cypher, we have 72 =: 12 X 6, and 576 = 8 X 72. Hence the operation may be performed thus: — Multiplicand 821436 Multiply by 6 in the 6th place. 4928616 Previous product X 12 ... 59143392 for 72 thousands. Second product X 8 473147136 for 576 units. Same product as before 552478139136 : three lines saved. Other modes of contraction will appear as we proceed. Sbct. V. Division of Whole Numbers. Division is a rule by which we determine how often one number is contained in another. Or, it is a rule by which, when we know a product and one of the factors which produced it, we can find the other. The number to be divided in called the dividend ; that by which it CHAP. I.] ]>I VISION OP WHOLB NUMBBRB. 11 is diTided, the divisor; and that which results from the divisioD, the ([ua^tnt. When division and multiplication are regarded as reciprocal operations, the dividend is equivalent to the product^ the divisor is equiTalent to the tntdtiplier^ and the quotient is equivalent to the mtd- tipUeand. Rule. — Draw a curved line both on the right and left of the dividend, and place the divisor on the left ; then find the number of times the dirisor is contained in as many of the left-hand figures of the dividend as are jost necessary, and place that number on the right. Multiply the divisor by that number, and place the product under the above- mentioned figures of the dividend. Subtract the said product from that part of the dividend under which it stands, and bring down the Bext figure of the dividend to the right of the remainder. Divide the remainder thus increased, as before ; and if at any time it be found less than the divisor, put a cypher in the quotient, bring down the next figure of the dividend, and continue the process till the whole is finished : the figures thus arranged will be the quotient required. Examples, Divide 743256 by 324. Dividend. Divisor 324)743256(2294 Quotient. 648 Divisor 324 Quotient 2294 648 3045 2916 Pro 131)135076(1031^^^: ' 131 1296 2916 648 648 1296 1296 Remain 71)29754(419^ 284 ►of 743256 In these two ex% amples the num- bers which re- main are placed ot^ their respect- ive divisors, and "attached to the quotients ; the meaning of which will be expbiined when we treat of 135 71 644 639 5 Remain. 407 393 146 131 15 Remain. Note.-^Wheu the divisor does not exceed 12, the operation may readily be perfoimed in a single line; as will appear very evident if 12 DIVISION OF WHOLE NUMBERS. [PABT I. tbc following example be compared with the two methods of working the first example in multiplication. Divide 38616 by 8. 8)38616(4827 Dividend 38616 32 Divisor 8 66 64 Quotient 4827 21 16 56 56 Here 38 contains 8 four times, leaving a remainder of 6; these carried as 6 tens to the next 6, make 66, which contains 8 eight times, leaving 2, which carried as 2 tens to the next figure 1, make 21 : and so of the rest. In division, also, upon the same principle as in multiplication, the la- bour may often be abridged by taking component parts of the divisor. Thus, in the first example, the divisor is equal to 4 times 81, or 4 times 9 times 9. Hence the dividend may be divided by 4, 9, and 9, successively, as in the margin, and the result will be the same as before. Divide 743256 by 4 this quotient 185814 by 9 and this 20646 by 9 Quotient 2294 Since 25 is a fourth part of 100, and 125 the 8th part of 1000, it will be easy to multiply or to divide by either of these numbers iu a single line — thus. To multiply 4827 by 25, put two cyphers on the right, which is equivalent to multiplying by 100; and divide by 4. 4)482700 120675 Answer. To divide 582100 by 25, strike ofi^ two figures on the right hand, which is equivalent to dividing by 1 00 ; then multiply by 4. 5821 100 4 To multiply 6218 by 125, put 3 cyphers, which is equivalent to multiplying by 1000; then di- vide by 8. 8)6218000 777250 Answer. To divide 4567000 by 125, strike off three figures on the right hand, which is equivalent to dividing by 1000; then multiply by 8. 45671000 8 23284 Answer. 36536 Answer. 2758 2758 3099 3099 4C9 1029 469 1029 7355 CHAP. I.] VULGAR FRACTIONS. 13 PROOF OF THE FIRST FOUR RULES OF ARITHMETIC. Simple as these four rules are, it is not unusual to commit errors in working them : it is, therefore, useful to possess modes of proof. 1. Now, addition may be proved by adding downwards, as well as upwards, and observing whether the two sums agree ; or, by dividing the numbers, to be added into two portions, finding the sum of each, and then the sum of those two separate amounts. Thus, in the margin, ^ss^ 5857 the sum of the four numbers is 7355 ; the sum of the two upper ones 5857, 1498 of the two lower ones 1498, and their sum is 7355, the same as before. 7355 2. The proof of subtraction is effected by adding the remainder to the subtrahend ; if their sum agrees with the minuend the work is right, otherwise not. 3. Multiplication and division reciprocally prove each other. There is also another proof for multiplication, known technically by the phrase casting out the nines. Add together the numbers from left to right in the multiplicand, dropping 9 whenever the sum exceeds 9, and carry on the rem£under, dropping the nines as often as the amount is beyond them ; and note the last remainder. Do the same i^-ith the multiplier and with the product ; then multiply the first two remainders and cast the nines out of their product; if the remainder is equal to the last remainder, this is regarded as a test that the work is right. Thus, taking the second example in multiplication, the figures in the multiplicand amount to 6 above two nines, those in the multiplier to 6 above three nines, those in the product to above six nines ; the product 6 x G of the two first excesses is 36, or above four nines : the coincidence of the two O's is the proof. It is plain, however, that the proof will be precisely the same so long as the figures in the product be the same, whatever be their order : the proof, therefore, though ingenious, is defective •. A similar proof applies to division. Sect. VI. Vulgar Fractions. The fractions of which we have already spoken in Sect. L, are usually denominated Common or Vulgar Fractions, to distinguish * The correctness of this proof, with the exception above specified, may be shown algebraicaUy, thus : — put M and JV — the number of nines in the mul- tiplicand and multiplier respectively, m and n their excesses ; then, 9 M •\- m =^ the multiplicand, and 9 Jv -f- n » the multiplier, and the product of those factors will l>e — 81 M N + 9 Af n + 9 iV m = m n ; but the three first terms are each a precise number of nines ; because one of the factors in each is so ; these, therefore, being neglected, there remains m n to be divided by nine ; but m n is the product of the two former excesses : therefore the truth of the method is evident. Q> E. D. 14 VULGAR FRACTIONS. [PART I. them from another kind, hereafter to he mentioned, called DecinuU Fractions, A fraction is an expression for the value of any part of an integer, or whole numher, such numher heing considered as unity. Thus, if a pound sterling he the unit, then a shilling will he the twentieth part of that unit, and /our pence will he four twelfths of that twen- tieth part. These represented according to the usual notation of vulgar fractions, will he -^^ and -^-^ of -^^ respectively. The lower numher of a fraction thus represented (denoting the numher of parts into which the integer is supposed to he divided) is called the denominator ; and the upper figure (which indicates the numher of those parts expressed by the fraction) the numerator. Thus, in the fractions ^, -j^, 7 and 15 are denominators^ 5 and 8 numerators. Vulgar fractions are divided into proper, improper, mixed, simple, compound, and complex. Proper fractions have their numerators less than their denomi- nators, as ^, ^, &c. Improper fractions have their numerators equal to, or greater than, their denominators, as ^, i^, &c. Mixed fractions, or numbers, are those compounded of whole numbers and fractions, as 7^9 12^, &c. Simple fractions are expressions for parts of whole numbers, as %, ^ &c. Compound fractions are expressions for the parts of given fractions, as I- of I, ^ of -3?^, &c. Cimijiex fractions have either one or both terms fractional 52 12 6^ numbers, as ^^ j^, ^^ &c. The value of a fraction is not altered by multiplying or dividing both its numerator and denominator by the same number; thus i? \'> W^ Jli' *^® ^ equal, although successively multiplied by 2, 6, and 12. Any number which will divide two or more numbers without remainder, is called their common measure. REDUCTION OF VULGAR FRACTIONS. This consists principally in changing them into a more commodious form for the operations of addition, subtraction, &c. Case I. — To reduce fractions to their lowest terms, Rtde, — Divide the numerator and denominator of a fraction by any number that will divide them both, without a remainder ; the quotient again, if possible, by any other number : and so on, till 1 is the greatest divisor. Thus, m^ =z If* = ^\\ = i\ = I, where 5, 3, 7, 7, re- spectively, are the divisors. ^"■5 IHi = h ^7 dividing at once by 735. CHAP. I.] VULGAR FRACTIONS. 15 Nnfte, — This number 735 is called the greatest common measure of the terms of the fraction : it is found thus — Divide the greater of the two numbers by the less; the last divisor by the last remainder, and so, on till nothing remains : the last divisor is the greatest common measure required*. Case II. — To redu4X an improper f ration to its equivalent whole or mixed number. Rule, — Divide the numerator by the denominator, and the quotient will be the answer: as is evident from the nature of division. EtX, — Let ^^ and ^4^ be reduced to their equivalent whole or mixed numbers. 43)957(22^} Answer. 274)5480(20 Answer. 86 548 97 86 = 11 Cask III. — To reduce a mixed number to its equivalent improper Jraction ; or a whole number to an equivalent fraction having any assigned denominator. Rule, — This is, evidently, the reverse of Case II.; therefore multiply the whole number by the denominator of the fraction, and add the numerator to obtain the numerator of the fraction required. Ex, — Reduce 22|:^ to an improper fraction, and 20 to a fraction whose denominator shall be 274. (22 X 43) + 11 = 957 new numerator, and \^ the 1st fraction. 20 X 274 z= 5480 new numerator, and ^^ the 2nd fraction. * The foUowing theorems are useful for abbreviating Vulgar Fractions :-— Theorems. 1. If the last digit of any number be divisible by 2, the whole number is divisible by 2. If the two last dig:its be divisible by 4, the whole number is divisible by 4. If the three last digiu be divisible by 8, the whole number is divisible by 8. And, generally, if the last n digits of any number be divisible by 2*, the whole number is divisible by 2*^. 2. If a number terminate with 5, it is divisible by 5 ; and if it terminate in 0, it is divisible by either 10 or o. 3. If the sum of the digits constituting any number be divisible by 3 or 9, the whole is divisible by 3 or 9 ; and if also the last digit be even, the whole number is divisible by 18. 4. If the sum of the digits constituting any number be divisible by 6, and the right.hand digit by 2, the whole is divisible by 6 : for by the data it is divisible both by 2 and 3. 5. If the sum of the 1st, 3rd, 5th, &c , digits constituting any number be equal to that of the 2nd, 4th, 6th, Ac., that number is divisible by 11 : for if a, 6, c, (f , e, m, n, be the digits, constituting any number, its digits, when multiplied by 11, will become (8) (7) (6) (5) (4) (3) (2) (1) «, a+4, A+c, c+rf, rf-h^, c-fm, m-^n^ n ; where the odd temns are » to the even. id VULGAR FRACTIONS. [pART I. Case IV. — To reduce a compound frartion to an equivalent simple one. Rule, — Multiply all the numerators together for the numerator, and all the denominators together for the denominator, of the simple fraction required. If part of the compound fraction be a mixed or a whole number, reduce the former to an improper fraction, and make the latter a fraction bv placing 1 under the numerator. When like factors are found in the numerators and denominators, cancel them both. Ex,— Reduce |? of \ of ^ of ^ of -,8^ to a simple fraction. 2x3xr> x7x8 2x5x8 1x6x8 1x5x4 20 3X4X7X9 Xll""-! X 9 X ll""2 X 9 X ll""! X 9 X ll^OQ Here the 3 nnd 7 common to numerator and denominator are first cancelled ; then the fraction is divided by 2 ; and then by 2 again. /t^x.— Reduce throe farthings to the fraction of a pound sterling. A farthing is the fourth of a penny, a penny the twelfth of a shilling, nnd a shilling the twentieth of a pound. Therefore ^ of -f^y of ^ zz ^J^ = -^^^y the answer. 2^ Ex, — Simplify the complex fraction ... H Here, reducing the mixed numbers to improper fractions, we have 8 — : multiplying by 3, to get quit of the denominator of the upper V 8 fraction, we have p: multiplying by 5, to get quit of the denominator of the lower fraction, we have ^J: dividing both terms of this fraction by 8, there results ^ for the simple fraction required. Case V. — To reduce fraction% of different detioininators to equi- valent fractions having a common detiominaior. Rule. — Multiply each numerator into all the denominators except its own, for new numerators ; and all the denominators together for a common denominator. Ex, — Reduce |, ^, and §, to equivalent fractions having a common denominator. 2 X 7 X 9 = 126 \ 6 X 3 X 9 = 162 [ the numerators. 5 X 3 X 7 = 105 ) 3 X 7 X 9 = 189, the common denominator. Hence the fractions are -[|^, -^, ^-g^, or *«, * J, ^^, when divided by 3. Hence, also, it appears that f exceed §, and that 4 exceed ^, Ex, — Reduce | of a penny, and J of a shilling, each to the fraction of a pound ; and then reduce the two to fractions having a common denominator. * of a penny = ^ of ^^ of 4^ = -p*,^^ = ^^^ of a pound. % of a shilling = ^ of ^^ = ^% = ^ = ^ of a pound. Hence ^ of a shilling are 1 times as much as | of a penny. CBiP. I.] VULOAB FRACTIONS. 17 Mute.— Other methods of reduction will occur to the student after tolenble practice, and still more after the principles of algebra are ttqaired. ADDITION AND SUBTRACTION OP FRACTIONS. RuU,—lf the fractions have a common denominator, add or sub- tract the numerators, and place the sum or difference as a new numc- ntor oTer the common denominator. If the fractions have not a common denominator, they must be re- duced to that state before the operation is performed. In addition of mixed numbers, it is usually best to take the sum of tbemtegere, and that of the fractions, separately ; and then their sum, for the resolt required. Examples. 1. Find the sum of f , ^, and |. . l;f- ^ t* "T-P + H + fi = V»' = 2H- z. Take | of a shilhng from y^ of a pound sterhng. I of a shilling = | of ^>„ = ^^ of a pound = -^q. Also -rV of a pound = /^j. Hence ^% - -^^^ = ^^Vff = ^»j = 11 pence. 3. Find the difference between 12 J and 8 J. »2| -H = V - V = W - Vo" = SV = 4/»- MULTIPLICATION AND DIVISION OF FRACTIONS. R*ik 1. To multiply a fraction by a whole number, mtdtiplif the KMPKrttfor by that number, and retain the denominator : — Or, divide ^dtnomnator by the same number, (if a multiple of it,) and retain tbe numerator. 2. To divide a fraction by a whole number, multiply the denomi- ■•tor by that number, and retain the numerator : — Or, divide the MJWrotor by the same number, (if a multiple of it,) and retain the Nominator. 3. To multiply two or more fractions is the same as to take a friction of a fraction ; and is, therefore, effected by taking the pro- duct of the numerators for a new numerator, and of the denomina- tors for a new denominator. (The product is evidently smaller tlian eitber factor when each is less than unity.) *. To divide one fraction by another, invert the divisor, and pro- «wd as in moltiplication. (The quotient is always greater than the dividend when the divisor is less than unity.) Examples. 1. Multiply { by 2, and divide ^ hy 5. c 18 DECIMALS. [PABT I. Multiply 25 by f , and divide f by ji 1-i, Ans, 3 =8 ^2;andf --^=^ l-V = Afultiply £2 IBs. U. bv 3}, and divide £4 15«. by 3^, £2 13«. 4</. = 2 + IJ + T*/ of A = 21 = f, and = 8 X J = V == «/ = 9^ = £9 6«. 8</. £4 158. H- 3i = 4| -r- 3i = V -^ V = V liJ = £1 8«. 6</. i^Tofe. — In the multiplication of mixed numbers, it is often less laborious to iV perform the multiplication of each part separately, and collect their sum, as in the margin, than to reduce the mixed numbers to improper fractions, and re- duce their product back again to a mixed number. 45 45 8 T 45 17 Multiply 45f ByJTf 7 =315 1 ten 3= 45 . I = .30 % = 'jn Product 808^ Sect. VII. Decimal Fractions, The embarrassment and loss of time occasioned by the com- putation of quantities expressed in vulgar or ordinary fractions, have inspired the idea of fixing the denominator so as to know what it is without actually expressing it. Hence originate two dispositions of numbers, decimal fractions and complex numbers. Of the latter, such, for example, as when we express lineal measures in yards, in feet (or thirds of a yard), and inches (or twelfths of a foot), we shall treat in the following section, and shall here confine ourselves to the former. Decimal fractions, or substantively, decimals^ are fractions always having some power of ten for their denominator ; but for the sake of brevity only the numerator is expressed, being written as tL^whok number with a dot placed on its left hand, which dot determines the value of the denominator, the number of cyphers in the denominator being always equal to the number of figures to the right of the dot, or as it is termed decimal point ; if the number of significant figures in the numerator is not sufficient, cyphers are added to the left hand. It is evident that the values of decimals decrease in the same tenfold proportion from the point to^^-ards the right hand, as those of integers mcrease towards the left : — thus Igai r fraction, tV is written •1 >i 99 aha 99 •01 » 99 TO (TU 99 •001 >i 99 TTJOcJi) 99 •0001 »> 99 iV >9 •7 » 99 tVo »> •43 » 99 "rioinr »> •0125 i« written •3 * >i •6 4 y» •428571 m >f •29504 CHAP. I.] DECIMALS. 19 The vulgar fraction, 7-/^ is written 7-3 42^*5 „ 42-85 57,VoV >> 57-217 &c. &c. The Talae of a decimal fraction is not altered by cyphers on the Jight hand : for '500, or -^W^, is in value the same as ^^, or -5, that is |. When decimals terminate after a certain number of figures, they are called/iii^«,as-125 = Vt5V(jpi-958 = T^V« = m- ^ . . When one or more figures in the decimal become repeated, it is oaHed a repeating or circulating decimal ; and a dot is placed over the figure to be repeated, if only one, or if more than one, over the first and last figures ; thus : — •333333, &c. '666666, &c. •428571428571, &c. •29504504, &c. When the circulating portion of the decimal is preceded by other figures which do not circulate, (as in the last example,) it is called a mixed circulate. Rules for the management of this latter kind of decimals are given by several authors ; but, in general, it is more simple and commodious to perform the requisite operations by means of the equivalent vulgar fractions, the method of obtaining which is given in Case III. RBDUCTION OP DECIMALS. Reduction of Decimals is a rule by which the known parts of given integers are converted into equivalent decimals, and vice verad. Case I. — To reduce a given vulgar fraction to an equivalent dedmoL Rule. — Annex as many cyphers to the numerator as may be neces- sury, then divide by the denominator, and point off in the quotient as many places of decimals as the number of cyphers added to the numerator; if the quotient does not contain so many figures, the deficiency must be made up by cyphers placed on the left hand. JSxamplea. 1. Reduce ^, |, i^, ^, to equivalent decimals. 2) 1-0 4) 3-00 •5 decimal = | ; *75 decimal = ^ : C 4)70000 ( 8)6000 16} «4 ^ 4)1-7500 I 8)^75000 •4375 decimal =- -f^; '09375 decimal = ^ •asHB - c 2 20 DECIMALS. [PABT I. 2. Reduce ^\ and ^i to equivalent decimals. / 3)4-000000 27 ( 9)1*333333 •148148, &c. = •148 decimal z= ^\ i 7)11-0000000 63' I 9) 15714285714285 •1746031746031, &c = •174603* decimal = i^. These two are evidently circulating decimals, in the former of which the figures 148 become indefinitely repeated, in the latter the figures 174603. 3. Reduce 14«. 6d, to the decimal of a pound. First, 14*. ed. = JJ + i of ^ig = IS + ViJ = 18- Then J 8 — 1^" = ''^^^' '^® decimal required. 4. Reduce tJ^ to its equivalent decimal. 57) 44-000000 (-77192, &c. decimal = ^. 399 170 114 56 Note, — The above fraction is = J x |^, of which the two denomi- nators arc both jortm^ numbers^ (that is, divisible by no other number than unity,) the entire equivalent decimal is a circulate of 18 places, t. e. one less than the last prime .... -771929824561403508, 7719, &c. over again ad infinitum*, • There are many curioui properties of fractions i « 'i 42867, Ac whose denominators are prime niimliers, one of I ^ •285714 An which may lie here shown in reference to fractions * "* . ' .» ^' having the denominator 7- The circulating fi^nires ? ~ •428671, Ac of the equivalent decimals are precisely the same, ^ "" -671428, Ac for \y ), &C., and in the same order: the cir- & ^ *714286 Ac culate merely oommeooes at a different place for each Z a.. , .<:* . numerator. * * 867142, Ac CHIP. I.] DECIMALS. 21 Gin II. — Antf decimal being given to find its equivalent vulgar fnttm; or to express its value bg integers of lower denominafions, i^.— When the equivalent vdgar fraction is required, place Qoder the decimal as a denominator a unit with as many cyphers as there are figures in the proposed decimal ; and let the fraction so con- ititnted be reduced to its lowest terms. Or, if the value of the decimal he required in lower denominations, Doltiplj the given decimal hy the number of parts in the next less deDomioation contained in its integer; and point off, from right to left, aa many figures of the product as there were places in the given decimal. Multiply the decimal last pointed off by the value of its integer, in the next inferior order, pointing off the same number of dednuJs as before : and thus continue the process to the lowest bteger, or until the decimals cut off become all cyphers ; then will tkeseTenJ numbers on the left of the separating points, together with the remaining decimal, if any, express the required value of the given decimal. Examples, 1. Find the vulgar fractions equivalent to '25 and '375. •25 = ^V^ = i ; and '375 = ^Vrf^^ = |, Answers. 2. Fbd the value in shillings, &c. of '528125 of a pound. •528125 20 10-562500 >Ans. lOs. Bid 12 \ 6-7500 = 6| 3. Fmd the value of *74375 of an acre. •74375 4 2-97500 > Ans. 2 roods 39 perches. 40 ^ 39000 Cm III. — To reduce a circulating decimal to its equivalent vtdgar fi^ttum, BtJs, — Take the figures in the decimal and place them as a whole Biimber for the numerator, and under them for a denominator as many S*! aa there are figures in the circulate, and the fraction thus formed will be equivalent in value to the given decimal. If the decimal is a mixed drcukte, subtract the finite part (or the %iii«s which are not repeated) from the whole mixed circulate, (both 22 DECIMALS. [part I. considered as whole numbers,) for the numerator; and for the deno- minator, take as many 9's as there arc figures in the circulating por- tion of the decimal, with a8 many cyphers to the right as there are figures in the finite portion of the same. Examples. The circulate 3 = i i •06 = ^ = h » -549 — wJ *^ ^ » 7630 = "rm = 7A The wwerf circulate -6409 = ^^ll^o* = il* 5-29504= 5»^|J-J^«5' = 5iH ADDITION AND SUBTRACTION OF DICIMALS. These operations are performed precisely as in whole numbers, the figures being so arranged that units stand under units, tens under tens, &c., or, ^which amounts to the same thing,) so that the decimal points stand under one another. Thus, 421-75 From 24861 78 Add } 32-8165 Take 14-56789 together i -0027 11- Remains 2471*60511 Sum 465-5692 Proof 2486-17300 MULTIPLICATION AND DIVISION OP DECIMALS. Here, again, the operations are performed as in integers : Then, in multiplication, let the product contain as many decimal places as there are in both the multiplier and multiplicand, cyphers being pre- fixed, if necessary, to make that number ; and, in division, point off as many decimals in the quotient as the number in the dividend ^including the cyphers supplied, if there be any) exceeds that in the oivisor. Examples. Multiply 43-7 by 39 1, and 2 4542 by -0053. 43-7^ 391 ' ^Here 437 x 391 ^^-}88S^ = 170^*0%. as in the decimal i operation. 2-4542 •0053 437 3933 . 1311 73626 122710 •01300726 170-867 Here one cypher is prefixed to make the requisite num- ber of decimals in the product. CBIP. I.] COMPLEX NUMBBBS. 23 2. Divide 172*8 by -144, and 192 by 5*423. •144) 172-8 ( 1200- quotient. 5423 ) 192000 ( 35-40475 144 16269 288 29310 288 27115 00 21950 = 21692 In the first of these examples, the two 25800 cjphen brought down, together with the deci- 21692 mil 8, make the number of decimals in the diridend the same as in the divisor, there- 41080 fore the quotient is composed entirely of in- 37961 tegers. In the second example, 3, the num- ber of decimal places in the divisor, taken 31190 from 8, the nnmber in the dividend (including 271 15 thoK brought down), leave 5 for the decimtd places in the qnotient. 4075 Skt. VIII. Complex Fractions used in the Arts and Commerce. In the arts and in commerce, it is customary to assume a series of nnits bsTing a constant relation to each other, so that the units of ODe denomination become fractions of another. One farthing, for eumple, is ^ of a penny, 1 penny -/.t of a shilling, 1 shilling ^*jy of a poand, or ^y of a guinea. One lineal inch, again, is yV of a foot, 1 foot i of A yard ; and so on, according to the relations expressed in the tible« in Chap. V. on Mensuration, Sect. I. The arithmetical opera- tioDs on complex numbers of these kinds are usually effected by limpler rules than those which apply to vulgar fractions generally ; of which it will, therefore, be proper here to specify a few. BBDUCTION. Here we have two general cases : Ca81 I. — When the numbers are to be reduced from a higher de- nomination to a lower : — Bide. — Multiply the number in the higher denomination by as ■uy of the next lower as make an integer, or one, in that higher, <id set down the product. To this product add the number, if any, vhich was in this lower denomination before ; and multiply the sum hy as Buny of the next lower denomination as make an integer in tht present one. Proceed in the same manner through all the dcnomi- B*^s to the lowest, and the number last found will be the value 24 COMPLEX NUMBERS. [part I. of all tlic numbers whicli were in the higher denominations taken together. Case IL — fV/ten the numbers are to be reduced from a lower de- nomination to a higher : — Rule.—DWidc the given number by as many of that denomination as make one of the next higher, and set down what remains. Divide the quotient by as many of this as make one of the next higher de- nomination, and set down what remains in like manner as before. Proceed in the same manner through all the denominations to the highest; and the Quotient last found, togetlier with the several re- mainders, if any, will be of the same value as the first number proposed. The method of proof is to work the question back again. Examples, 1. Reduce <£l4 to shillings, pence, and farthings; and 24316 farthings into pounds, &c. 14 4)24316 20 280 shillings 12 12) 6079 pence 20) 506 7 3360 pence 4 £25 6«. 7d. 13440 farthings 2. Reduce 22 Ac. 3 R. 24 P. into perches; and 52187 perches into acres. a. r. p. 22 3 24 40)52187 4 91 roods 4)1304 27 40 Ac. 326 R. 27 P. 3664 perches. ADDITION. /2w/e.— Place the quantities to be added so that those of the same denomination may be all under each other. Then add up the numbers in the right hand column, and divide their sum by the number of units of that denomination contained in an unit of the next denomination to the left, write the remainder (if any). at the foot of the first column, and carry the quotient on to the addition of the second column^ and thus proceed until all have been added up. CHIP. I.] COMPLEX NUMBSBS. 25 Examj^es. £ «. dL lb. oar. duft. gr, tb, om. dujt.gr, '368 10 3 / 14 6 12 13 / 10 8 11 17 1257 10 5 i 17 5 3 12 t 42 5 16 12 .jJ 88 U 4i . ,, J 15 9 16 . , J 12 2 14 18 ^^^j 33 10 ^"^"^i 2 7 15 20 ^"^"^ \ 51 6 22 f 12 13 5 f 13 2 10 19 f 24 9 17 17 8 8 8^ \ 4 1 5 21 V 29 4 18 22 Sum 769 4 2 Sam Q^ 11 18 5 Sum 171 2 12 SUBTRACTION. Ruk, — Write the smaller number beneath the greater, taking care to keep the same denominations under each other. Then begin at the right band and subtract the lower number of each denomination from the upper, writing the remainder underneath. When the lower number of any denomination is greater than the upper one of the same, add to the latter as many units as are con- tamed in one unit of the next greater denomination, always taking care, when such has been done, to add one to the next lower number to the left. Examples, £. M. d. £. g, d, lb, OM,dtoLgr. From 16 12 8f From 21 13 4J From 18 9 10 8 Take 10 11 sl Take 18 9 8| Take 9 10 15 20 Bern. 6 1 2| Rem. 3 3 8| Rem. 8 10 14 12 MULTIPLICATION. i^ti/e.— Place the multiplier under the lowest denomination of the multiplicand. --Multiply the number in the lowest denomination by the moltiplier, and find how many integers of the next higher de- nomiDation are contained in the product, and write down what remains. — Carry the integers, thus found, to the produce of the next higher denomination, with which proceed as before ; and so on, thioagh all the denominations to the highest ; and this product, together with the several remainders, taken as one number, will be the whole amount required. If the multiplier exceed 12, multiply successively by its component puis; as in the following examples :•— 26 OOMPLEX NUMBKBS. [PABT I;. ExamfleM, £ 9m d, a. r. p. 1. Mo]tiplj4 17 e\ by 441, and 3 2 14 by 531. 10 £ $, d. 4 17 Q\ 85 3 20 9 X 7 X 7 = 441 9 10 43 17 lOj 7 307 4 7 An*. £2150 U lOj 358 3 for 100 5 1793 3 for 500 107 2 20 3 times 10 3 2 14 1 top line. ^n«. 1904 3 34 DIVISION. Bide. — Place tbe divisor and dividend as in simple division. — Begin at the left hand, or highest denomination of the dividend, which divide by the divisor, and write down the quotient. — If there be any re- mainder after this division, find how many integers of the next lower denomination it is equal to, and add them to the number, if any, which stands in that denomination. — Divide this number, so found, by the divisor, and write the quotient under its proper denomination. — Proceed in the same manner through all the denominations to the lowest, and the whole quotient, thus found, will be the answer required. £ B. d. 2. Divide 521 18 6 by 432. 432 = 12 X 12 X 3. Therefore, by short division : 12)521 18 6 12)43 9 10^ 3)3 12 6i + I a farthing. Qtwiient £} 4 1| + f of a farthing. CHAP. I. DUODECIMALS. 27 By loDg division : — £ 9. d. £ 9. d. 432)521 18 6 (1 4 1 J + f of a farthing. 432 89 20 432)1798(4 1728 70 12 432)846(1 432 414 4 432)1656(3 1296 m^a^i 360 DUODECIMALS. Fmetions whose denominators are multiples of 12, as 144, 1728, &e., ire called duodecimals; and the division and suh-di vision of the iateger sre Hudersiood without being expressed, as in decimals. The method of operating by this class of fractions is principally in use >DiODg artificers, in computing the contents of work, of which the ^eDsions are taken in/eet, inches^ and twelfths of an inch. Rule. — Set down the two dimensions to be multiplied together, one under the other, so that feet shall stand under feet, inches under iocbes, &c. Multiply each term in the multiplicand, beginning at the West, by the feet in the multiplier, and set the result of each iou&ediately under its corresponding term, observing to carry 1 for vvery 12, from the inches to the feet. In like manner, multiply all tbe multiplicand by the inches of the multiplier, and then by the twelfth parts, setting the result of each term one place removed to tbe right hand when the multiplier is inches, and two places when the parts become the multiplier. The sum of these successive products will be the answer reqmred. Or, instead of multiplying by the inches, &c., take such parts of the multiplicand as these are of a foot. 28 P0WBB8 AND ROOTS. [PABT 1. Examples. 1. Multiply \2ft. 7i ins. by 1 /t. 3 J tVw. //. in$. ' 12 7 4 or, 7 3 9 88 3 4 3 1 10 9 5 6 92 2 7 6 tnM. 3 == i of 1 ft. 9' = I of 3 ins. fL %n», ' 12 7 4 7 ft. 88 3 4 3 I 10 9 5^ 92 2 7| 2. Multiply Z5fi. ^\ins. into 12^. S^ins. ft. ins. ' 35 4 6 or, 12 3 4 tnM. 3 = ^ of 1 ft. 4' = ^ of 3 ins. 424 6 8 10 1 11 9 6 6 434 3 11 ft. ins. ' 35 4 6 12 ft. 424 6 8 10 n 11 9i 434 3 _n The feet in the answers are square feet, but the numbers standing in the place of inches are not square inches but twelfth parts of square feet, each part being equal to 1 2 square inches ; and the numbers in the third place being twelfth parts of these are square inches : in like manner, if the operation be carried further, every successive place will be a twelfth part of that preceding it. Sect. IX. Powers and Roots. • A power is a quantity produced by mttltipl}'ing any given number, called the root or radix, a certain number of times successively by itself. The operation of thus raising powers is called involution. Thus, if 3 is the root, 3 = 3 is the Ist power of 3. 3x3=3'= 9, is the 2d power, or square of 3. 3 X 3 X 3 = 3 '= 27, is the 3d power, or cube of 3. 3 X 3 X 3 X 3 = 3^ = 81, do. 4th power, or biquadraU of 3, &c. &c. &c. CHAP. I.] BVOLUTION. 29 L- ,., Table of thejirsi Nine Patters of the first Nine Numbers. lit A 2d 1 4 9 16 3d 1 8 4di &th 6tk 7tli &th dUi 1 1 1 1 1 1 16 81 32 64 128 356 61t 243 729 2187 6fi61 65536 19683 64 G2& 1296 1024 3125 4096 1S625 16384 £^144 3S 1 1S5 78126 279936 390625 1679616 1953125 m 64 216 7776 4G6&G 10077696 343 240] 512 4006 16B07 117649 823543 2097152 5764801 — ^ — 16777216 4035364^7 32768 262144 134317728 tl 729 6561 ^9049 531441 47629C^ 43046731 367420480 So again, J x f = $ = square off;|x| = ^ = cube of 3 > ^ X J- = ^, biqoadrate of 4 ; and so of others. Where it is eyident, that while the powers of integers become successively larger ^nd larger, the powers of pure or proper fractions become sue- oessively smaller and smaller. EVOLUTION. Evdulumy or the extraction of roots, is the reverse of involution. Any power of a given number may be found exactly ; but we cannot, conversely, find every root of a given number exactly*. Thus, we know the square root of 4 exactly, being 2 ; but we cannot assign exactly the cube root of 4. So, again, though we know the cttbe root of 8, viz. 2, we cannot exactly assign the square root of 8. But, of 64 we can assign both the square root and the cube root, the former being 8, the latter 4. By means of decimals we can in all cases approximate to the root to any proposed degree of exactness. Those roots which only approximate are called surd roots, or surdSy or irrational numbers ; as v^2, V5, \/9> &c., while those which can be found exactly are called rational; asx/9 = 3, i/125 = 5, iyie == 2. 1. — To extract the square root. Rule, — Divide the given number into periods of two figures each, by setting a point avei' the place of unitSy another over the place of * For the method of extracting roots by logarithm!, see page 47* dO EVOLUTION. [pari I. hundreds, and so on over every second figure, both to the left hand in integers, and to the right hand in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of a quotient figure in division. Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period, for a dividend. Double the root above-mentioned for a divisor ; and find how often it is contained in the said dividend, exclusive of its right- hand figure ; and set that quotient figure both in the quotient aad divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to it the next period of the given number, for a new dividend. Repeat the same process, viz., find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before; and so on through all the periods, to the last*. Note. — The best way of doubling the root, to form the new divisors, is by adding the last figure always to the last divisor, as ap- pears in the following Examples. — Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of cyphers, two in each period. Examples. 1. Find the square root of 17*3056. 17*3056(4'16 the root : in which the number of IG decimal places is the same as the number of decimal periods into which the given number was divided. 81 130 1 81 826 4956 6 4956 * The reason for separating the figures of the dividend into periods or portions of two places each, is, that the square of any single figure never con- sists of more than two places ; the square of a number of two figures of not more than four places, and so on. So that there will be as many figures in the root as the g^iven number contains periods so divided or parted off. And the reason of the several steps in the operation appears from the algebraic form of the square of any number of terms, whether two or three, or more. Thus, 36* » 30* + 2 . 30 . 5 + 6% or genmilly (a -f 6)* = a* + 2 a 4 -|- i» = a* 4* (2 a 4* 6) ^t the square of two terms ; where it appears that a is the first term of the root, and b the second term ; also a the first divisor, and the new divisor is 2 a 4* 6, or double the first term increased by the second. And hence the manner of extraction is as in the rule. CHAP. I.] EVOLUTION. 31 2. !Rnd the square root of 2, to six decimals. 2(1-414213 root. 1 24 1 4 00 96 281 1 400 281 2824 4 11900 11296 28^82 2 60400 56564, 28284 1 383600 1 282841 2828423 10075900 8485269 1590631 3. Find the square root of -f-^. -r\ = '4>16666666y &c. 6-4i 6666(0-64549, &c. 36 124 566 4 496 1285 5 7066 6425 12904 64166 4 51616 12908 9 1255066 1161801 93265 ^ote. — In cases where the square roots of all the integers up to 1000 are tabulated, such an example as the above may be done more easily by a little reduction. ThusV^j =\/(t^^ X ^f) =x/ -j^^ = ^^60 = ?:?^ = -645497, &c. 2. — To extract cube and higher rooU. The rules usually given in books of arithmetic for the cube and higher roots, are very tedious in practice: on which account it is advisable to work either by means of approximating rules, or by means of logarithms*. The latter is, generally speaking, the best method. We shall merely present here Dr. Hutton's approximating rale for the cube root. Rule, — By trials take the nearest rational cube to the given num- ber, whether it be greater or less, and call it the assumed cube. Then say, by the Rule of Three, as the sum of the given number and double the assumed cube, is to the sum of the assumed cube and double the given number, so is the root of the assumed cube, to the root required, nearly. Or, as the first sum is to the difference of the given and assumed cube, so is the assumed root, to the difference of the roots, nearly. • See page 47- 32 PROPOBTION. [part I. Again, by using, in like manner, the cube of tbe root last found as a new assumed cube, another root will be obtained still nearer. And so on as far as we please ; using always the cube of the last found root, for the assumed cube. Example, To find the cube root of 210358. Here we soon find that the root lies between 20 and 30, and then between 27 and 28. Taking therefore 27, its cube is 19683, which is the assumed cube. Then 19683 2 21035-8 2 39366 21035-8 42071-6 19683 As 60401-8 : 61754-6 27 4322822 1235092 27 : 27-6047 60401-8)1667374-2(27-6047 the root nearly. 459338 Again, assuming 27*6 36525 and working as before, the 284 root will be found to be 42 27-60491. Sect. X. Proportion. Two magnitudes may be compared under two different points of view, that is to say, either by inquiring what is the excess of one above the other, or hoto often one is contained in the other. The result of this comparison is obtained by subtraction in the first case, by division in the second, the quotient resulting being termed the ratio of the two numbers. Thus 3 maybe regarded as the ratio of 12 to 4, since *3p or 3 is the quotient of the numbers 12 and 4. The first of two numbers constituting a ratio is called the antece- dent^ the second the consequent. The difference of two numbers is not changed by adding one and the same number to each, or by subtracting the same number from each. Thus 12 - 5 = (12 + 2) - (5 + 2) = 14 - 7 = (12 - 2) - (5 - 2) = 10 — 3. In like manner, a ra^io is not changed by either muUiplging both its terms, or dividing both its terms by the same number. Thus V =(y X !) = !« =c/-f)=f CHIP. I.] PROPOBTION. 33 F/lnaliiy of differences^ or equidifference^ is a term used to indicate tlattbeMerence between two numbers is the same as the difference between two other numbers. Such, for example, asl2 — 9 = 8 — £gva/i<jf of ratios^ or proportion^ is similarly employed to denote that tbe ratio of t^wo numbers is the same as that between two others. TbusSO and 10, 14 and 7, have 2 for the measure of the ratio : we kie therefore a proportion between 20 and 10, 14 and 7, which is tbos expressed, 20 : 10 : : 14 : 7, and thus read 20 are to 10 a« 14 are to 7. The same proportion may also be represented thus, ^g = y . Tboagb, \>y whatever notation it be represented, it is best to read or eoamer&te it as above. It is true, however, that in all cases when tvo fractions are equal, the numerator of one of them is to its de- QOffiinator, as the numerator of the other is to its denominator. In a proportion, as 20 : 10 : : 14 : 7, the second and third terms we called the meanSy the first and fourth the extremes. Wben the two means are equal, the proportion is said to be con- tinued. Thus 3 : 6 : : 6 : 1 2 are in continued proportion. This is osually expressed thus -rr 3 : 6 : 1 2 ; and the second term is called the mean proportional. In the case of equidifference^ as 1 2 — 9 =7 — 4, the sum of the extremes (12 + 4) is equal to that of the means (9 -r 7). In like JMnnerin a proportion, as 20 : 10 :: 14 : 7, the product of the ex- tremes (20 X 7) is equal to that of the means (10 x 14). The converse of this likewise obtains, that if 20 x 7 = 10 X 14, then 20:10:: 14 : 7. Hence, 1. If there be four numbers, 5, 3, 15, 9, such that the products 5 X 9 and 3 x 15 are found equal, we may infer the equality of tbcir ratios, or the proportion ^ = ^ , or 5 : 3 : : 15 : 9. So that a proportion may always be constituted with the factors of two equal products. 2. If the means are equal, their product becomes a square ; there- fore the mean proportional between two numbers is equal to the square root of their product. Thus, between 4 and 9 the mean proportional i8%/(4 X 9) = 6. 3. If a proportion contain an unknown term, such, for example, as 5:3:: 15: the unknown quantity; since 5 times the unknown <IttDtity roust be equal to 3 x 15 or 45, that quantity itself is equal to 43 -r 5 or 9. Or generally, one of the extremes is equal to the prodact of the means divided by the other extreme ; and one of the OKans is equal to the product of the extremes divided by the other mean. 4. We may, without affecting the correctness of a proportion, object the several terms which compose it to all the changes which ciQ be made, while the product of the extremes remains equal to that ofthemeans. Thu8,for5 : 3 : : 15 : 9, which gives 5 x 9 = 3 x 15, we may I. Change the places of the means without changing those of the ^^Etremea, or change the places of the extremes without changing those of the means : this is denoted by the term aUernando, D 84 BULK OF THREE. [^A&T I. Thus, 5 : 3 : : 15 : 9 become 5 : 15 : : 3:9 or 9 : 3 : : 15 : 5 or 9 : 15 :: 3 : 5 II. Put the extremes in the places of the means ; this is oalled invertendo ; as 3 : 5 : : 9 : 15 III. Multiply or divide the two antecedents or the two consequents by the same number. It also appears, with regard to proportions, that the sum or the dif- ference of the antecedents is to that of the consequents, as either ante- cedent is to its consequent. And, that the sum of the antecedents is to their difference, as the sum of the consequents is to their difference. Tu ^ 3; 15 , ,, , 5 -h 15 5^15 ^^"^ "3"T^ = 1^ = v> and ^-^ —rz^' If there be a series of equal ratios represented by ^ = ijp = y == 6 ^ 10 + 14 -f 30 3g, we shall have 3 ^ ^ ^ ^ ^. ^^ = JJ = | = V> = &c. Therefore, in a series of equal ratios, the sum of the antecedents is to the sum of the consequents, as any one antecedent is to its consequent. If there be two proportions, as 30 : 15 : : 6 : 3, and 2 : 3 : : 4 : 6, then multiplying them term by term, we shall have 30 x 2:15 x 3 : : 6 x 4:3 x 6, which is evidently a proportion, because 30 X 2x3x6 = 15x3x6x4 = 1080. Thus, also, any powers of quantities in proportion are in proportion; and conversely of the roots. Thus, If 2 : 3 : : 6 : 9 then 2* : 3' : : 6'-* : 9' 2 : 3 : : 6 : 9 „ \/2 : -^3 : : -v/6 : v 9 2 : 3 : : 6 : 9 „ 2^ : 3' : : 6^ : 9' RULE OP THREE. When the elements of a problem may be so disposed that they form a proportion of which the quantity sought is the last term ; that is, when the first bears the same proportion to the second as the third does to the fourth or unknown quantity, its value may be easily determine<l, and the problem is said to belong to the Goklen EuUj or Bute of Three. Etdle. — Of the three given terms set down that which is of the same kind as the number sought, then consider from the nature of the problem whether this number will be greater or less than the term so put down ; if greater^ write on its right hand the greater, if leu^ the lesser, of the two remaining terms, and place the other on its left hand. Then multiply the second and third terms together, and divide their product by the first; the quotient will be the number sought It CEAP. I.] BULB OF THR£B. 85 moat be observed, that the first and third terms must be reduced to tlie nine deDomiiiation ; and if the second term is a compound nomber, it should be reduced to the lowest name mentioned ; unless the ibird term is a composite number, in which case it is generally better to multiply the second term (without any previous reduction) h the componeDt parts of the third, as in compound multiplication, ifler which divide the compound product by the first term, or by its &ctoi8. The answer will be of the same denomination as the second tern. Examples, 1. If 3 gallouB of brandy cost 2. How much brandy may be 19i.,what will 126 gallons cost bought for 39/. 18«., at the rate of It the same rate ? { 3 gallons for 1 9 shillings ? fA, t. gal. ! 3:19:: 126 : ? 19 1134 126 3)2394(798 sbillings 21 — or 39/. 18«. Ans. 29 ' 27 24 24 3. If 21 yards of cloth cost 2*^ 10#., what will 1 60 yards cost ? frfi. £ «. yds. Hefe,21:24 10:: 160 : ? 4 4x4x 10=160 98 4 19 gaL : 3 : 18 ? 21 392 10 (3)3920 £ 39 20 798*. 3 19) 2394 ( 126 ^wj. 49 38 114 114 4. If by selling cloth at 1/. 2«. per yard, 10 per cent, is gained, what would be gained if it had been sold at 1/. 5b. per yard ? £ t. £ 9. Here, 1 2: 110:: I 5 : ? 20 20 22 25 no 2750 22 [7)1306 13 4 £186 13<. 4d. Ana. (2) 111 )1375 Amount £ 1 25 Deduct 100 Gain per cent. £ 25 = d2 3^ aru -:f thru. [PABT I. of 5'».'. :-:-r 5 Te^rsw s: 4 £ ^' £ Here. H» : 4 : : o^M 4 l<Ni 2244? per ^. If 100 workmen can finish a piece of work in 1 2 days, how manj men working equally hard would hare finbhed it in 3 davs ? </. ir. d. 12: 100:: 3 : ? 12 3)1200 Aftneer 400 workmen. £22-4 2*1 80 Interest for I vear, £22 $# Then 1 : 22 S : : 5 5 ;fll2 rt^u^rvr. A distinct rule is usually given for the working of problems in Compound Frojxtrtion ; but they may generally be solved with greater mental facility by means of separate statings. Thus : — 7. If a person travel 300 miles in 10 days of 12 hours each, in how many days of 1 6* hours each may he travel 600 miles ? First, if the days ^ ere of the same lenj^h, it would be, bv simple proportion, m. d, m. As 300 : 10 : : 600 : 20 days. But these would be days of 12 hours each, instead of 16, of which fewer will be required. Hence, again, by simple propor- tion, h. d. h, d. As 12 : 20 :: 16 : 15 So that the answer is 1 5 days. 8. If a family of 9 persons spend 480/. in 8 months, how much will serve a family (living upon the same scale) of 24 per- sons 16 months? I First, as 9 : 480 P- 24:o£l280. But this would only be the expense for 8 months. Hence, again. m. £ As 8 : 1280 171. £ As 8 : 1280 : : 16 : 2560, tbe expense of the 24 persons for 1 6 months. yote. — The Rule of Three receives its application in questions of Interest^ Discount^ Fellowskipy Bartery &c. CHAP. I.] BBTXRMINATION OF RATIOS. 37 BBTERMINATION OP RATIOS. To find the ratio of two numberSy A and B, to each other. Rule 1. — DiTide JB hy A, then, unit^ or 1 : the quotient, will be tbe iim ratio ; and if the quotient be an integral number, the ratio will be expressed in the least terms possible. If, however, it contain a fiMOoD, proceed as follows : — first write, 1 : integral portion of Ike quUient -f the numerator of the fraction ; then, add 1 to the integral, and subtract the numerator of the fraction from the de- nominator, and write under the former, 1 : integral portion 4-1 — tk 's.ofihe numerator and denominator. Then, if the numerator and this difference bo nearly equal, add the two ratios together ; but if the numerator and difference are not nearly equal, divide the greater by the less ; then, multiply that ratio ending with either the numerator or the difference, whichever was the divisor, by the integral portion of this last quotient, and add to it the other ratio. With the three ratios thus obtained proceed in the same manner, from whichever two have the numbers appended by the signs -|- and — , nearest equal, to obtain a fourth ratio, and thus continue, until this appended number has been eliminated. Note. — The ratios thus found will be alternately greater and less than the true one, but continually approaching nearer to it. And that is the nearest in small numbers, which is immediately followed bj much larger numbers : the excess or defect of any one is equal to a fraction, having the number appended to that ratio by the sign + or — for its numerator, and the denominator belonging to the first quotieDt, for its denominator. Example \, To find the ratio of 10000 {A) to 7854 {B) in small numbers. 7854 H- 10000 =0t7^«5^ Then, Ist I. 2Dd 3rd II. 4th III. 5tb IV. 6th V. 7th 8th VI. 8th VII. 10th VIII. nth 1 : + 7854 or ratio of 1 to the integer and num. 1 : 1—2146 or integer 4-1— the '^ of num. and den. 2146)7854(3 d : 8—6438 or 2nd ratio x by 3. 4: 3 + 1416 or 1st and 3rd ratios added together. 5 : 4— 730 or 2nd „ 4th „ „ 9 : 7+ 686 or 4th „ 5th „ „ 14: 11— 44 or 5th „ 6th „ „ 44)686(15 210 : 165— 660 or 7th ratio x by 15. 219 : 172+ 26 or 6th and 8th ratios added together. 233: 183— 18 or 7th „ 9th „ 452: 355+ 8 or 9th „ 10th „ „ 8)18(2 38 DETERMINATION OP RATIOS. [PABT IX. X. 12th Idth Uth 15th 904 : 710-f 16 or llth ratio x by 2. 1137 : 893— 2 or 1 0th and 1 2th ratios added together. 2)8(4 4548 : 3572— 8 or 13th ratio x by 4. 5000 : 3927 -f or llth and 14th ratios added together. The ratios are numbered according to their convergence, with Roman numerals to the left hand, and are as follows : — ± A Ji 17 3 > T) A ±Jl AX9 AAA AAJL 7> 1I> 112 9 183> 366> OJLl 5000 853 > 3 92 T ' Of these the nearest in small numbers is -J^, differing from the true ratio by only j-^^Q-Qt and is indicated (as alluded to in the preceding note) by being immediately followed by the much larger numbers f^^. Example 2. To find the ratio of 268*8 to 282 in the least numbers. 2688) 2820 (l-,VftS 2688 Then, I. l8t 2nd II. 3rd 4th III. IV. 5th eth 7th V. 8th 9tb 1 I 19 20 40 41 61 183 224 132 : 1 -h 132 or ratio of 1 to the integer and num. : 2—2556 or integer -|- 1— the '^ of num. and den. 132)2556(19 : 19 + 2508 or 1st ratio x by 19. : 21 — 48 or 2nd and 3rd ratios added together. 48)132(2 : 42— 96 or 4th ratio x by 2. : 43+ 36 or 1st and 5 th ratios added together. 1 2 or 4th „ 6th ,, „ : 64— 12)36(3 : 192— : 235 36 or 7th ratio x by 3. or 6th and 8th ratios added together. Therefore the several ratios are -J-, ?% -J^, J|, and |j^. And the excess or defect of any one is seen by inspection; thus, -*^ differs from the true ratio only -jj^; and -|^, but -^11-^. Rule 2. — Divide the greater number by the less, and the divisor by the remainder, and the last divisor by the last remainder, and so on till remain. Then, 1 divided by the first quotient, gives the first ratio : And the terms of the first ratio multiplied by the second quotient, and 1 added to the denominator, give the second ratio : And in general the terms of any ratio, multiplied by the next quo- tient, and the terms of the foregoing ratio added, give the next suc- ceeding ratio. CHA?.I.] DBTSIUCINATION OV &ATIOe* 39 Example 3. LetthenoTObcrs be 10000 and 31416, or the ratio Hf?#- 10000)31416(3 30000 1416)10000(7 9913 88)1416(16 88 536 528 Tben, 8)88(11 88 -< s= Ist or least ratio. 17 7 7 i Y 7 — and = — = 2nd ratio. 8 ^ 21 21 ^- 1 22 7 112 , 112 + 1 113 Q , ,. __Y ift— and • — = s=3rd ratio. 22''^^'" 862 352 4-3 355 lL^n-'^^ and i?l£±:L = l?^=4tli ratio. S65 "^ ^^ " 3905 3905 + 22 3927 JEfzample 4. The mtio of 268*8 to 282 is required. 2688)2820(1 2688 132)2688(20 264 48)132(2 96 86)48(1 86 12)36(3 36 40 LOGARITHMIC ARITHMETIC. [PABT Then, - := 1st ratio. - X 20=- and ^^^^ = ^^ = 2nd ratio. 20 ^ 40 , 40 4- 1 41 « , . 21 ^ ^=42^"^^ IT^l = 43 =^r<l-^- 41 , 41 ,41+20 61 , ^ . ■TT- X 1 = — and 7- — - = -- = 4th ratio. 43 43 43 + 21 64 61 „ 183 ^ 183 +41 224 ,^ . gj X 3 = — and j^^-^^ = — = 5th ratio. Sect. XI. Logarithmic Arithmetic. As the nature and properties of logarithms are described in a sub- sequent part of this work, being so placed because such description could not have been properly understood without a certain acquaint- ance with algebra, we shall here only explain the use of the tables given in the Appendix, and the method of employing logarithms to facilitate the common processes of arithmetic. By an inspection of Tabic II., which contains the logarithms of all numbers from 1 to 100, it will be seen that each logarithm consists of two distinct parts, separated by a decimal point ; thus, the loga- rithm of 13 is 1*113943; the number to the left of the decimal point (or 1 in the above example), is called the index or charac- teristic*^ and its value depends only upon the number of digits in the quantity whose logarithm it is, without any regard to thet7a/tt€ of that quantity, and it is always 1 less than that number of digits ; thus, in the example, the characteristic of the logarithm of 13, which contains two digits, is 1, or one less than that number ; and it will be seen from the Table, that 1 is the characteristic of all the logarithms from 10 to 99, but that, for numbers below 10, the index is 0, and for 100 is 2, in each case 1 less than the number of digits in the quantity of whose logarithm it is the characteristic. The characteristic, there- fore, of the logarithms of all numbers equal to or greater than 1 and less than 10 is 0* 10 » 100 „ 1- 100 >» 1000 „ 2- 1000 >» 10000 „ 3- 10000 »> 100000 „ 4- &c. &c. &c. * In order to avoid confusion from the use of the leord index to signify two thincB, we shall throughout this work employ the term characterisHc when speaking of logarithms, and iruies when speaking of roots or powers. CHAP. I.] LOGARITHMIC ARITHMETIC. 41 When the qaantity is less than unity, the characteristic of its loga- rithm becomes negative, and its value is determined hy the number of cyphers which occur between the decimal point and the first signi- ficant figure, (the fraction being decimally expressed), and is always 1 greater than such n amber of cyphers; or it is equal to the differ- ence in the number of figures in the numerator and decimal denomi- nator; thas, the characteristic of the logarithm of •1 or iV is 1-^ •01 » lio »» 2- •001 » 16^6 >» 3- 0001 &c. >» 1 ?» 4. 10000 &c. &c. The decimal part of the logarithm, or that lying to the right of the decimal point, depends entirely on the relative value of the figures composing the quantity whose logarithm it is, and not at all upon the tttoal numerical value of that quantity ; thus, in the example already giTen, the decimal part of the logarithm of 13 is '113943, which is iko Ae decimal part of the logarithm of 1*3, or 130, or 1300, for in each case the 1 and the 3 have the same relative value. So that the decimal portion of a logarithm is always the same for the same fignrea, and is not altered by the addition of any number of cyphers either to the right or to the left hand of those figures, or what is equivalent, by the multiplication or division of the quantity by 10, or any power of 10 ; it is only the characteristic of the logarithm which alters its value, 1 being akded to the characteristic for every 10 hy which the quantity is mvltipliedy or subtracted from it for every 10 hy which the quantity is (/«t7M/e(/. Thus, the logarithm of 745800 being 5872622 that of 74580 is 4-872622 „ 7458 >» 3-872G22 745-8 >> 2-872622 74-58 »> 1-872622 7-458 » 0*872622 •7458 99 1-872622 „ 07458 >» 2-872622 •00745fi ^ » 3-872622 It must be borne in mind, that in the logarithm of a fractional qnantity, it is only the characteristic which has a negative value, and that the decimal pal^t of a logarithm is always positive. It is, how- CTer, sometimes convenient to have the whole logarithm expressed negatively, both characteristic and decimal ; for which purpose, sub- • The negative sign ( — ) ia always placed above the characteriatic, thus 2, initcad of before it, in order to avoid its being misunderstood for the sign of mbcractiim. 42 LOGARITHMIC ARITHMETIC. [PART I. tract the last right hand figure in the decimal portion from 10, and all the others from 9, and the result will he what is termed the arithmetical complement of the decimal, to which prefix the former characteristic less I, and the result will he a negative logarithm, equivalent in value to the original logarithm having only a negative characteristic; for example, the logarithm of '07458, as above, is 2^-872622, which is equivalent to — 1 127378. It is also frequently convenient to take the arithmetical complement of the whole logarithm, and this is obtained by subtracting the right hand figure of the decimal from 10, and all the others from 9, including the characteristic when positive^ but if negative it must be added to 9. Thus, the arithmetical complement of 3-14G128 is 6-853872 „ 207G276 „ 11-923714 „ 5-322839 „ 4-677161 „ i-986772 „ 10013228 USB OP THE TABLES. To find the logarithm of any given number. If the number is less than 100, its logarithm will be found in Table II., with its proper characteristic prefixed ; but if the number contains more than two figures, its logarithm may be found from Table III, as follows: — If there are only three figures in the num- ber, look for that number in the first column of the table, and on the same line in the next column to the right, under 0, will be found the decimal portion of the required logarithm, to which the proper cha- racteristic must be prefixed, according to the rules which we have just explained. If the quantity contains four figures, look for the first three figures in the first column as before, and the four last figures of the logarithm of the required number will be found on the same line with those three figures, and in that column which has at its head the fourth figure of the given number ; the two first figures of the logarithm will be found in the second column (headed 0), and which figures being common to all the logarithms enclosed by each pair of horizontal lines, it is unnecessary to repeat. Where these first figures change their value in the middle of a line, the same is indicated by a break in the horizontal line, thus, 139879 | 0194, which shows that the two first figures (13) have changed to 14, and the right hand logarithm is therefore 140194. The heading figures are repeated at every tenth line in the body of the tables, in order to facilitate their use. Examples. Required the logarithm of 734. In Table III., on the same line with 734 and under 0,. are found 5696, the four last figures of the logarithm^ to which the common CHIP. I.] LOOARtTHMIC ABITBlfBTIC. 48 figures 86 and the proper characteristic 2 being prefixed, we obtain 2'^569C, the logarithm required. Find the logarithm of 3476. Hpre, on the same line with 347 and under 6, will be found 1080, which, with the two first figures and the characteristic prefixed, is 3'54108O, the logarithm required. The log. of 5-84 is ^766413 „ 0932 „ 2-969416 10-24 „ 1 010300 „ 3708 „ 3-569140. When the quantity whose logarithm is required contains more than four figures, proceed as follows: — Find the logarithm for the first four figures as above, then look in the first column of Table I. for the first figures, and on the same line in the column having at its head the fifth figure will be found the quantity which must bo added to the logarithm already taken out^ to give the logarithm of the quantity first required. If the first four figures are not found in the first column of the table, then take the line containing the next less number to it. If the number whose logarithm is required contains more than ^yc figures, proceed as above to obtain the logarithm of the first five figures, then, on the same line of Table I. that the number added to the logarithm for the fifth figure was found, and in the column htTJDg at its head the sixth figure, will be found a quantity, which, dirided by 10 (or what is the same, having its right hand fisure taken twtj*), and added to the logarithm already found, will give the iogtfithm of the first six figures; again, on the same line and in the cokmn baring at its head the seventh figure, will be (bund a quan- tit?, which, divided by 100 (or baring two figures cut off from the right band), and added, will give the logarithm for seven figures t. Examples. Required the logarithms of 11488, 621547, 768654, 7642179. log. of the first four figures from Tab. Ill = 4059942 From Tab. I. on line with 1148 and under 8 ... = 302 Logarithm of 11488, as required = 4060244 Log.of6215 =1= 5-793441 From Tab. I. on line with 6160, the next less) _ „« No. in the tab. to 62)5 and under 4 J "" On aame line under 7 = 4 Logarithm of 621547 = 5793474 * If the figure thus cot off ezoeeds five, one must be added to the first right kind fijnire left. t See remark at page 45, with regard to the number of places to be de- 44 LOOABITHMIC ARITHMETIC. [PART I. Log. of 7686 = 5-885700 From Tab. I. on same line with 7686 under 5... = 28 On same line under 4 := 2 Logarithm of 768654 = 58 85730 Log. of 7642 = 3-883207 From Tab. I. on same line with 7552 under 1 ... = 5 On same line under 7 = 3 On same line under 9 = 7 99 513 Logarithm of 76421 79 = 3883217 To find the number answering to any given logarithm. Look in Table III. for the given logarithm, or the next less in value to it that can be found, then on the same line, in the first column, will be found the first three figures, and at the head of the column in which the logarithm was found, the fourth figure of the number sought. If the given logarithm is found exactly in the table, the figures thus obtained will be the required number, care being taken to point off one more figure to the left hand than there are units in the characteristic of the given logarithm, cyphers being at- tached to the right hand of the number, if requisite. If, however, the given logarithm is not found exactly in the table, subtract from it the next less logarithm found, calling the remainder the first differ- ence ; then look in Table I. on the same line with the four figures already obtained from Table III. (or the next less figures which can be found) for this difference, and at the head of the column in which it is found will be the fifth figure of the number sought. If the first difference is not found exactly in the table, look for the next less number to it, which subtract from the first difference for the second difference ; then add a cypher to this second difference, and look for it on the same line of Table I. as before, and the figure at the head of the column containing the nearest number to it, either greater or less, will be the sixth figure of the number required. Examples. Required the number answering to the logarithm 3*241756. Given log. = 3'241 756 Next less log. in Tab. III. =3*241546 = the log. of 1744* 210 first dif. In Tab. I. on same line with ) i nn • r j • i o o the next less No. to 1 7*4 2^ '^ *^"""^ '" *="'• * ]^ 110 second dif. On the same line 99 is found in col. 4 *04 The No. required = 1744*84 CHAP. 1.] LOGARITHMIC ARITHMETIC. 45 In this example the next less logarithm which can he found in Table III. is 3*241546, the numher answering to which, 1744, is the first four figures of the namher sought ; then subtracting this loga- rithm from the given logarithm, we obtain for the first difference 210, and looking in Table I. on a line with 1740 (the next less number to 1744), for the next less number to 210, we find 199, at tbe head of the column containing which is 8, the fifth figure re- quired; then subtracting 199 from 210, we obtain the second differ- ence, 11, and adding a cjrpher, the nearest number which we find on the same line is 99, at the head of the column containing which is 4, the sixth figure required. Required the numbers answering to the following logarithms : — 3510009, 2-475771, 5-871624. The number answering to the logarithm 3*510009 is found at oQcetobe32d6. Given log. = 2-476771 Next less log. = 2-475671 = the log. of 299-0 From Tab. I.... 100 = 1st dif. 87 •06 130 = 2nd dif. 130 -009 No. required = ■ = 5-871624 = 5-871573 = the log. of 299069 Given log. Next less log. 744000' Prom Tab. I.... 51 = 1st dif. 46 80- 60 62 9- The No. required = 744089- It should be observed here, that the number of figures which may ^ depended upon in any result obtained by logarithms, will be equal Jo the number of decimal places in the logarithms employed ; thus, in Qiing the tables appended to this work, the results obtained will ^ accurate to six figures, except towards the end of the tables, in which only five figures should be trusted. MULTIPLICATION AND DIVISION BY LOGARITHMS. To mulHpiy two numbers together, add together their logarithms, 4i6 LOOABITHIfIC ARITHMETIC. [PART I. and the sum will be the logarithm of their product; or, to divide one number by another, subtract the logarithm of the divisor from the logarithm of the dividend, and the remainder will be the log- arithm of the quotient of the two numbers. Ex, — Multiply 80 X 43 X 72*64; and divide the product of (7143 X 6278) by 3145. Add Log. 80 = 1-47712J; Log. 7143 log. 43 = 1-633468 add log. 6278 Jog. 72-54 = 1-860578 Log. 93576-4 = 4-971167 sub. log. 314-5 : 3'85S881 3-797821 7-651702 2-497621 Log. 142589 ;= 5154081 rROPORTION OR THE RULE OP THRBB BY LOGARITHMS. The Rule of Three is very readily performed with the aid of log- arithms, by simply adding together the logarithms of the 2nd and 3rd terms, and subtracting the logarithm of the 1st, the remainder being the logarithm of the 4th term, or number required to be found. Or, instead of subtracting the logarithm of the 1st term, we may €idd its complement, (the method of obtaining which has been already explained at page 42,) and subtract 10 from the character- istic of the result, which will, as before, be the logarithm of the 4th term. Examples. The following are the same as the 1st, 3rd, 4th, and 5th Ex- amples given in the Rule of Three, at pages 35 and 36. 1st Ex. Log. 19 = 1-278754 + log. 126 = 2-100370 3-3791-24 - log. 3 = 0-4771 21 3rd Ex. Log. 245 = 1-389166 4- log. 160 = 2-204120 3-593286 - log. 21 = 1-322219 Log. 798 =s 2-902003 Log. 186-667 = 2-271067 4th Ex. 5th Ex. Comp. of log. 22 = 8-657577 Comp. of log. 100 = 8000000 -f- log. 110 = 2041393 + log. 4 = 0602060 + log. 25 = 1-397940 + log. 560 = 2748188 Log. 125 = 2096910 Log. 22-4 = 1-350248 CHIP. I.J LOGABITHMIC ARITHlfKTIC. 47 EVOLUTION AND INVOLUTION BY LOOABITHM8. To perfonn the operation of involution^ or the raising of powers, it is only necessary to multiply the number, any power of which is required, by the index of that power, and the product will be the logirithm of the required power; and, inversely, the operation of miuiioH^ or the extraction of roots, is performed by simply dividing the logarithm of the number by the index of the root required, the quotient ynl\ be the logarithm of the root Examples. Square 84, cube 13, and raise 7 to the sixth power. Log. 84 = 1-924279 x 2 = 3-848558 = 7056 = 84- log. 13 = 11 13943 X 3 = 3341829 = 2197 = 13» log. 7 = 0-846098 x 6 = 5070588 = 117649 = 7^ Extract the square root of 576, the cube root of 4913, and the axth root of 46656. Log. 576 = 2-760422 -r 2 = 1 3802 11 = 24 = n/ 576 log. 4913 = 3-691347 -^ 3 = 1-230449 = 17 = 'V 4913 log. 46656 = 4-668908 -r 6 = 0779151 = 6 = V46666 • It is necessary here to make a few remarks on performing the operations of evolution and involution on logarithms with negative cWacteristics. In doing this, it must be borne in mind that it is only the characteristic which has a negative value, the decimal part of tie logarithm being always positive ; therefore, if it is required to multiply a logarithm with a negative characteristic by any number, iim multiply the decimal part of the logarithm, pointing oflf as many decimal figures in the product as there were in the logarithm, then multiply the characteristic, and subtract from the product the num- ber (if any) pointed off to the left in the first product, the result will W the negative characteristic, and the decimals pointed off in the first product will be the decimal part of the required logarithm. of Examplei. Required the square of *25, the cube of '375, and the sixth power The logarithm of -25 = T-397904 First multiply 897940 by 2 2 _ _-795880 Then 1x2=2 2-795880 = -0625 = -25*. 48 LOGARITHMIC ARITHMETIC. [PART I. The logarithm of -375 is f-574031. •574031 3 1-722093 1 X 3 = 3- 2-722093 = -05273437 = •375». The logarithm of 7 is 1-845098 -845098 6 5-070588 1x6 = 6- 1-070588 =-117649 = •7«. To divide a logarithm with a negative characteristic by any nam- ber :— If the characteristic is a multiple of that number, that is, if k is divisible by it without remainder, proceed as in ordinary division ; if not, separate the characteristic from the decimal, and add to the characteritttic a number which will make it divisible^ and prefix to the decimal the same number, then divide both by the given divisor, and the quotients will be the characteristic and decimal of the logarithm required. Examples, Required the square root of -0625, the cube root of -74, and the fifth root of -543. The logarithm of -06*25 is 2-795880 Then 2)2-795880 r397940 = -26 = \/0625 The logarithm of -74 =1-869232. Then 1+2=3-7-3 = 1- and 2-860232 H- 3 = -956411 V-74 = -9045 = 1-956411 The logarithm of -543 is 1-734800 Then T -f 4=5-r-5 =1- and 4-734800 -f- 6 = -946960 V-543 = -885034 = 1946960 CRIP. 1.] PBOPRRTIRS OF NUMBERS. 49 Sbct. XII. Properties of Number m. To render these intelligible to the student, we shall here collect a few definitions. Def. I. A tintV, or unity ^ is the representation of any thing con- adered individoally, without regard to the parts of which it is com- 2. An integer is either a unit or an assemblage of nnits ; and a Jhetum is any part or parts of a unit. 3. A multiple of any number is that which contains it some exact namber of times. 4. One number is said to measure another, when it divides it with- out leaving any remainder. 5. And if a number exactly divides two, or more numbers, it is then called their common measure, 6. An even number, is that which can be halved, or divided into tvo equal parts. 7. An odd number, is that which cannot be halved, or which differs from an even number by unity. 8. A prime number, is that which can only be measured by 1 , or onitT. 9. One number is said to be prime to another when unity is the odv Dnmber by which they can both be measured. 10. A composite number, is that which can be measured by some Dumber greater than unity. 11. A perfect number, is that which is equal to the sum of all its difigore, or diquot parts : — thus 6= f + f + f • Pnp. 1. The sum or difference of any two even numbers is an tten number. 2. The sum or difference of any two odd numbers is even ; but the nm of three odd numbers is odd, 3. The sum of any eifen number of odd numbers is even ; but the nm of any odd number of odd numbers is odd, 4. The sum or difference of an even and an odd number is odd, 5. The product of any number of even numbers is even ; and any power of an even number is even, 6. The product of any number of odd numbers is odd; and every power of an odd number is odd, 7. The product of any number of even numbers, by any number of <^nombers, is even. 8. An odd number cannot be divided by an even number, without t remainder. 9. If an o</<ar number divides an even number, it will also divide the Wfofit 10. If a number consist of many parts, and each of those parts ^n a common divisor d, then will the whole number, taken col- ^▼ely, be divisible by d, 11. Neither the sum nor the difference of two fractions, which are n their lowest terms, and of which the denominator of the one con- E 50 PR0PBRT1B8 OF NUUBRR8. [PART T. tains a factor not common to the other, can be equal to an integral number. 12. If a square number be either multiplied or diTided by a square, the product or quotient is a square ; and conversely, if a square num- ber be either multiplied or divided by a number that is not a square, the product or quotient is not a square. 1 3. The product arising from two different prime numbers cannot be a square number. ] 4. The product of no two different numbers prime to each other can make a square, unless each of those numbers be a square. 15. The square root of an integral number, that is not a complete square, can neither be expressed by an integer nor by any rational fraction. IG. The cube root of an integer that is not a complete cube cannot be expressed by either an integer or a rational fraction. 17. Every prime number greater than 2, is of one of the forms 4n 4- 1, or 4» — 1. 18. Every prime number greater than 3, is of one of the forms 6n -r 1, or G w — 1. 1 9. No algebraical formula can contain prime numbers only. 20. The number of prime numbers is infinite. 21. The first twentv prime numbers ore 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 4f, 43, 47, 53, 59, 61, and 67. 22. A square number cannot terminate with an odd number of cyphers. 23. If a square number terminate with a 4, the last figure but one (towards the right hand) will be an even number. 24. If a square number terminate with 5, it will terminate with 25. 25. If a square number terminate with on odd digit, the last figure but one will be eren ; and if it terminate with any even digit, except 4, the last figure but one will be odd, 2G. No square number can terminate with two equal digits, ex- cept two cyphers or twoybwr*. 27. No number whose last, or right-hand digit is 2, 3, 7, or 8, is a square number. 28. If a cube number be divisible by 7, it is also divisible by the cube of 7. 29. The difference between any integral cube and its root is always divisible by 6. 30. Neither the sum nor the difference of two cubes can be a cube. 31. A cube number may end with any of the natural numbers 1, 2, 3, 4, 6, 6, 7, 8, 9 or 0. 32. If any series of numbers, beginning from 1, be in continued geometrical proportion, the 3rd, 5th, 7th, &c. will be squares ; the 4th, 7th, 1 0th, &c. cubes ; and the 7th, of course, both a square and a cube. 33. All the powers of any number that end with either 5 or 6, will end with 6 or 6, respectively. CHAP. 1.] PROPERTIES OF NUMBRRS. M U. Any power, w, of the natural numbers, 1, '2, 3, 4, 5, 0, &c. las as many orders of differences as there are units in the common exponent of all the numbers; and tlie last of those differences U a constant quantity, and equal to the continual product Ix*2x3x4x xw, continued till the last factor, or the number of factors be w, the exponent of the powers. Thus, Tbe 1st powers I, Si, 3, 4, 5, &c., have but one order of differences 1111 &c., and that difference is 1 . The iind powers I, 4, 9, 16, 25, &c., have two orders of (Ufferences 8 5 7 9 of which tbe last is constantly 2 = 1 x 2. The 3rd powers 1, 8, 27, 04, 125, &c., have three orders of differences 7 19 37 61 12 18 24 6 6 of which the last is 6 = 1 X 2 X 3. In like manner, the 4th, or last, differences of the 4th powers, •re each = 24 = 1 x 2 x 3 x 4 ; and the 5tb, or last differences of the 5th powers, are each 125 =: 1 x 2 x 3 x 4 x 5. 35. If unity be divided into any two unequal parts, the sum of the square of either of those parts added to the other is the same. Thosj of tbe two parts ^ and ♦, | + (f )« = ^ -h (\)- =: ^ ; so, •g«n, of the parts J and ^, ?- + (f)« = -J + {fY = if * For tbe demonstrations of these and a variety of other properties of nmnbers, those who wish to pursue this curious line of inquiry Bttjconsalt Legendre "Sur la Theorie des Nombres," the "Dis- <iaifltioDet Arithmeticse" of Gauss, or Barlow's " Elementary Inves- tigition of the Theory of Numbers." Alto, for the highly interesting properties of Circulating Decimals^ ttd their connexion with prime numbers^ consult the curious works rf the late Mr. H. Goodwyn, entitled " A First Centenary," and " A Ttble of the Circles arising from the Division of a Unit by all the htesenfrom 1 to 1024." K 2 52 DEFINITIONS [pART I. CHAP. II. ALGRBRA. Skct. I. Dejinitunis and Notation. Algebra is the science of the computation of magnitudes in gene- ral, as arithmetic is the particular science of the^ computation of numhers. Every figure or arithmetical character has a determinate and indi- vidual value ; the figure 5, for example, represents always one and the same number, namely, the collection of 5 units, of an order depend- ing upon the position and use of the figure itself. Algebraical cha- racters, on the contrary, must be, in general, independent of all par- ticular signification, and proper to represent all sorts of nambers or quantities, according to the nature of the questions to which we apply them. They should, moreover, be simple and easy to trace, so as to fatigue neither the attention nor the memory. These advantages are obtained by employing the letters of the alphabet, a, A, c, &c. to represent any kinds of magnitudes which become the subjects of mathematical research. The consequence is, that when we have resolved by a single algebraical computation all the problems of the same kind proposed, in the utmost generality of which they are sus- ceptible ; the application of the investigation to all particular cases requires no more than arithmetical operations. It is usual, though by no means absolutely necessary, to represent quantities that are known by the commencing letters of the alphabet, as a, ^, c, dy &c., and those that are unknottn by the concluding letters w, Xy t/y z. But it is often convenient, especially as it assists the memory, to represent any quantity which enters an investigation, whether known or unknown, by its initial letter ; as Bum by «, pro- dtict by py density by rf, velocity by c, time by t; and so of others. In addition to the signs already explained as being used in arith- metic, the following symbols and modes of expressing certain opera- tions, are employed in algebra. The product of two or more quantities is expressed either by in- terposing the sign of multiplication, as a x b x c x d; or by inter- posing dotSy which have the same signification, bs a . b . c .d; or, more simply, by placing the letters merely in juxtaposition, B&abcd. And as it is immaterial in what order the multiplication is performed, it is usual to write the letters after each other in the order in which they would stand in the alphabet, placing any figures which may occur in the product. THiP. 11.] AND NOTATION. 53 at the commencement. Thus, the continual product of /2 ^, a;, 7 e, lod S c, may be written either ^bzl eScy or x ^bl eSc^ &c., but it is preferable and usual to write them i^ b e e x. The figures % 3, 7 and 42, by which the letters are multiplied, are termed their mficienti; when a letter is not preceded by any figure (as x in the example), its coefficient is 1, or unity. We give the name term to any quantity separated from another by the sign -I- or — . When an algebraical expression consists of only 1 term, as 4 a it is called a monomial. 2 „ ac — 4 a & . . . „ binomial*. 8„ o + fc — a c . , ' „ trinomial. 4 „ c + </— Sy-H X . „ quadrinomial. {multinomial^ or polynomial . The signs + and — , which in arithmetic simply indicate the opera- tioos of addition and subtraction, are employed more extensiyely in algebra, to denote, besides addition and subtraction, any two opera- tioiis or any two states which are as opposed in their nature as addi- tkn tnd subtraction are. And if, in an algebraical process, the i^ + is prefixed to a quantity to mark that it exists in a certain itete, pontion, direction, &c., then, whenever the sign — occurs in eoDoexion with such quantity, it must indicate precisely the con- tiary state, position, &c. and no intermediate one. This is a matter of pare convention, and not of metaphysical reasoning. Other cha- rKters might have been contrived to denote this opposition ; but they would be superfluous, because the characters + and — , though ori- gioally restricted to denote addition and subtraction, may safely be otended to other purposes. ■ignifies any^ ^ theright, \ nifies \ to the left, (, forwards, J (.backwards. |. . / Increase, "^ — a signi- ( Decrease, ^^^ ^ J Money due, f respond- j Money owing, ^"^^°*^^ ( Motion upwards, ) mg v Motion doT^nward. And so on in every species of contrariety. And two such equal quan- tities connected togeUier in any case destroy each other's effect, or are «|8al to nothing, as + a » a'= 0. Thus, if a man has but 10/. ad at the same time owes 10/. he is worth nothing. And, if a ves- •d which would, otherwise, sail six miles an hour, be carried back a miles an hoar by a current, it makes no advance. * When the second term of a binomial has the negative sign ( — ) it is called 54 ADDITION AND SUBTRACTION. [PART T. Like Bigns are either all poMtive (-f ), or all negative ( — ). Aud unlike are when some are positive and others negative. If there be no sign before a quantity, the sign + is understood. Like quantities are such as contain the same letters, and differ only in their coefficients^ asa — 7 a, or9a6j:+ 2abx^^abx, Un- like quantities are such as contain different letters, as a 6 — c/ + bkx. When any number of terms are collected under a vinculum or bar, thus, a -f /» — c/^, or inclosed in parentheses thus, (a + 6 — c/)*, it denotes that the whole quantity so enclosed is to be taken col- lectively, and subjected to whatever operation or process is indicated by the symbol without the bar or parenthesis ; thus, in the example, that the compound quantity a -f 6 — cfy is to be squared. An equation is when two sets of quantities which make an equal aggregate are placed with the sign of equality ( = ) between them ; As 12 4- 5 = -40 — 3, or X -f ;/ = a + ft — c d. The quantities placed on both sides the sign of equality are called respectively the inemhcrs of the equation. The sign x indicates that the quantity before which it is placed is infnite^ or unlimited in its value. The symbol a , placed between two quantities, signifies that the second varies as the first : thus, a 7 /> is read a varies as b. The word therefore being of frequent occurrence in deducing the successive steps of algebraic processes, is denoted by . • . . Sect. II. Additimi and Subtraction, In algebra, the operations answering to those of addition and sub- traction in arithmetic cannot with propriety be called by those names, as either of them in algebra frequently involves the actual use of both those processes. They would be better expressed by some general term denoting the incorporation or striking a balance between the several quantities employed. When two simple quantities, or monomialsy are to be added to- gether, as 2 ft and c, or 4 a and 5 a, it is done by connecting them together with the sign of addition (-f); as2ft + c, 4a-|-6a; but when the quantities are similar, as in the second example, the expres- sion may be simplified by adding together the two coefficients, and subjoining the common letter, which would then become 9 a. In like manner, the subtraction of simple quantities, or monomialsy may be expressed by the sign of subtraction ( — ), or in the case of simple like quantities by actually subtracting tlie coefficients, and subjoining the common letter ; thus, the subtraction of ft from a, /from c, 4 ft c from 7 ft c, is performed thus, a — ft, c — /J 3 ft c. It not unfrequently occurs that the quantity to be subtracted is greater ihan the quantity from which it is to be taken, the consideration of which being totally different from all that the learner has been accus- tomed to in arithmetical operations, involves a difficulty in its appro- CBAP. IL] addition AND SUBTRACTION. 55 bension, bat this will soon be remoTed by an attentive perusal of the piragraph at page 53, explaining the sense in which the signs + and — are used in algebra. Where such occurs, the less quan- tity must be taken from Uie greater, and the negative sign prefixed to the remamder. To add together compound quantities, or polynomials, as (o + 2 6-1- c) and (4 o -f 5 c — rf), write one after the other, with their proper signs attached, omitting the parentheses, as fl + 2ft-fc + 4a 4- 5c— </, which may be simplified by the fur- ther addition or subtraction of its separate terms, and then becomes 5a + 26+ 6 c — </. Wben quantities to be added are presented promiscuously, it is best to classify them preyious to their addition. Thus, 3 aS — 3 * c, + 2 c2, + 4 ^, 3 a^ -h 5 6c - 2 c^ + 7aS +56c, -f aS — 2cS —46c, 7 a« —86c -f 4rf when arranged become as in the margin, a^ _ 4 6 c + 2 c^ and their sum is readily obtained, as in thefoorth line. 11 a« — 2 6c + 4 rf To subtract one compound quantity from another, change the sign of each term of the subtrahend^ and then proceed as in addition. Thus, 4 « 6 — 3 6 c And 4a6 — 3c«-f 6c — (2 a 6 — 6 6 c) «. ( a 6 — c^ — 2 6 c) become 4 a 6 — 3 6c become 4a6 — 3c*+ 6c — 2a6-f66c — tf6-f c'^+26c Resalt 2 a 6 -h 3 6 c Result 3a6 — 2c« + d6c The reason of changing the signs of the subtrahend may be ex- plained as follows : — Let it be required to subtract (c— cQ from (a -f 6) ; let us first subtract c, and the result will be a + 6 — c ; but it is evi- dent that in subtracting c from (a -f 6), we have taken away more than we ought, since it is only c— d which should have been sub- tncted, we have therefore taken away too much by </, and to obtain the correct result, must add it again to a + 6 — c, which then bc- eomes a + 6 — c + </> in which it is obvious that the signs of the nbtrahend (c — d) are changed. In addition and subtraction of algebraic fracti(m8^ the quantities Buist be reduced to a common denominator, and occasionally undergo other reductions similar to those in vulgar fractions in arithmetic; >Qd thee the sum or the difference of the numerators may be placed om the common denominator, as required. 56 MULTIPLICATION. [PART I. m, « . c ad be ad + be ..a b e a b a e be And, J-- 4.— +— ^-1- — 4.4.-.= o c ae a b c b a a2 b^ <r a""' b^^ a« c« b^c" aoc abc abc abe abc abe a« + *• + c« + a' ft" 4- a'^ c- + ft^ ^^ a ft c . , a c ad ^ bc^ a — iT ac + c^ arf — dx c cd c d ac -j- ex -^ ad + dx _ a{C'-d) -f ^ (c + rf) ed ~ ed And, *+* *-* ft — X ft 4- 0? (ft2 4- aft« + ^«) - (ft« - 2fta? 4- ««) 4 ft. ft- - a?2 "■ ii ^0.2' Sect. III. Multiplication. 1 . To multiply one monomial by another, multiply their coefficients for the coefficient of the product, and subjoin to it the letters com- posing the two quantities in their natural order, and if the same let- ter occurs in both, add together the two exponents for its exponent in the product, 1 being understood where no other exponent is ex- pressed. In attaching the proper sign to the product, observe that the product of two factors having contrary signs is negative^ and of two factors having the »ame sign positive. Or briefly, that like signs give 4-9 and unlike signs — . Note. — The general rule for the signs may be rendered evident from the following definition ; multiplication is the finding a magni- tude which has to the multiplicand the proportion of the multiplier to unity. Hence, the multiplier must be an abstract number, and, if a simple term, can have neither 4- nor — prefixed to its notation. Now first, 4- a X 4- »» = 4- »» «> for the quality of a cannot be altered by increasing or diminishing its value in any proportion; therefore the product is of the quality pltUy and m a by the definition is the product of a and m. Secondly, — ax 4-m = — mo, for the same reasons as before, mutatis mutandis. Thirdly, 4- a x — m has no meaning ; for m must be an abstract number, therefore here CHIP. II.] MULTIPLICATION. 57 we can baTe no proof. Bat + ax (m -^ n) s= ma ^ na^ n being Jess than m; for a taken as often as there are units in m is = m a by the first case ; bat a was to have been taken only as often as there ire units in m — n; therefore a has been taken too often by the units in n; consequently a taken n times, or n a, must be subtracted ; sod of course ma — n a is the true product. Fourthly, — ax (m — n)a=: — ma -f na. For -^ a x m s= — ma (by case 2); but this, as above, is too great by — na; therefore — ma with n a subtracted from it is the true product ; but this, by the rule of subtraction, is = — wi a + w a. 2. To find the product of two pdynomiahy multiply each term of the one mto all those of the other, following the rule given for mo- uomials. 3. To multiply algebraic fradionSy take the product of the nume- rators for the new numerator, and that of the denominators for the new denominator. Examples. 1. iah X bed =i ^ . 5 . ah .cd z= ^0 abed, 2. 8 a« ^» X 4 a* 6 = 8 . 4 . a2 . a* . 6' . * = 32 a« + * fc' + i = 32o» bK 3. Multiply 2o + *c-2*« By2a — *c + 2ft« 4 a« + 2 a * c — 4 a 5« — 2a*c— 6«c« +2 5«c + Aab^ +2 5»c — 4 6* Product, 4 a« — *« c« +4^«c--4 i^* U + fc l» -H 2 a 6 + ft* b o«+ ab a» +2a« b + ab* ab -{- b* + a«i^ + 2ad« + ft* a« + 3a6 + ft« a» •f3a«ft + 3aft« +6» a + ft ft ••{:i « +aft - a ft - ft« a« -ft« 58 DIVISION. [part I. a + b a ^b _ ( g -j- 6) (a — b) _ a" — b\ c rf"" c X d ^ c d 2« "dab 3ac__ IS a- be x __ 9 a j? *o c ^b " tiabe "l - — — - = 9aa?. iVb/tf. — From the above examples (4, 5, and 6) we may Icani — 1. Tliat the square of the sum of two quantities is equal to tbe sum of tbe squares of tbe two quantities together witb twice their pro- duct. 2. That tbe cube of tbe binomial a -^ by is a* -f S a' b + 'dal^ -{• b\ 3. That the product of tbe sum and difference of two quantities is equal to tbe difference of their squares. Sect. IV. Division, 1. To divide one monomial by another, divide the coefficient of the dividend by tbe cocfRcient of tbe divisor for tbe coefficient of the quotient, and subjoin to it a fraction having for its numerator tbe let- ters composing the dividend, and for its denominator those of the divisor, and if tbe same letter occurs in both, subtract tbe exponents for tbe exponent of tbe same letter in tbe quotient ; if the exponents as well as tbe letters are alike, both the quantities may be struck out. Tbe same rule applies to tlie signs in division as in multiplication. 2. To divide a polynomial by a monomial^ divide each term of the polynomial by the monomial according to Rule 1, and connect the results by their proper signs. 3. To divide two jwlynomials one by tbe other, arrange them witb respect to tbe powers of tbe same letter, then divide tbe first terms one by tbe other, and thence will result one term of tbe quotient ; multiply tbe divisor by this, and subtract the product from tbe divi- dend : proceed witb the remainder in tbe same manner. 4. To divide algebraic /ractiont^ invert the terms of tbe divisor, and proceed as in multiplication. Ea!amples, 1. 'ladH'c^Sab^ y^a*-'b'-'c=^id'bc. U. 15 a ♦ b' ^ 5 a- b' = »/ a^-'b'-' = 3 a b\ 8. 12 «^ — 2a- f -f c -r 2 rt- = a- — c 4- r^ • 2a^ H.r 12j:v l).r//^ i (i X -f 12 aw/ — .;? V ^ -t- '\ .v = h --— ^ -^^ = CHAP. II.] DIVISION. ,59 O 6. Divide af^ -^ Sjp^z -^ Sxsr — sf"* by x ^ z. x^z)x^-^SarZ'\-3xz^^z^(a^-'^xz'\-z^ quotient — 2ar^2r4- ^x^ xz'^z^ a? jy^ — 2r* Divide a* — b^ by a — b. a - 6) a» — 6Va* + a^ 6 + a* 6' + a ft* 4- ft* quotient. a^-aH a* ft -an* a'b''-a'b^ Here the second tenn of a^ ft'* the dividend is brought c^b^ ^ ab^ down to stand over the corresponding term in a ft* — ft* the last product. aft* — ft ^ 8. Divide 1 by 1 — a? \-^x)\ (1 + 0? + ;r- + ar^ + a^ + ^-^ 1 —a? ^""^ J? — X' -«' -«« *•- • ar^ ar"* 60 INVOLUTION. [PAKT I. 9. 2d?' a' H- X' ^x a -j- X a* -^ x^ U X' a + X _ 2 J?- (o + ;p) X "~ (a* + 0?*) ;p x'^h' 10. X' ■\- hx ^* — 6* F.X X — b x'^^hx'^h'' x--h {x^hf x{x^-h) ;p' + ft ' ft" a? (;r -h ft) (j? — ft) X {x- — ft-) ^ X ' 11. Divide 96 — 6 a* by 6 — 3a. Qiwt. 16 +8a -j-Aa^ -^Hal 12. Divide 10a' + 1 1 a'ft — 19 aftc — 15 a^c -(- 3 aft« + ISftc'' — 6 b'c by 3 a ft + 5 a- — 5 ft c. Quot, 2 a + ft — 3 c. 13. Divide x"- +y" + -, by a: -f y + - • QtioL a? — y -f- - . Sect. V. Involution, 1. To invoice or raise monomials to any proposed power. Rule, — Involve the coefficient to the power required, for a new coefficient. Multiply the index of each letter by the index of the required power. Place each product over its respective letter, and prefix the coefficient found as above : the result will be the power required. All the powers of an affirmative quantity will be -f : of a ne^ive quantity, the even powers, as the 2nd, 4th, 6th, &c., will be + ; the odd powers, as the 3rd, 5th, 7th, &c., will bo — . To involve fractions, apply these rules to both numerator and denominator. The application of these rules is fully exemplified in the following table of BOOTS AND POWBBS OP U0N0MIAL8. Root a -b V* e a a* I'x 2y 1 -a- a," Square <^ ¥ V*' a* a' 1 a'V *»«' Cube a" -ft' X «' a" 8y 1 -a"*' 4th Power a* V X/x' a* a" ley 1 a»*'^ 5th Power a" -b' Vx^ a' 32 y 1 - a*^x" CHiP. II.] INVOLUTION. 61 ^. To involve polyn&miali, M.'Mnldply the given (quantity into itaelf as many times, wanting one, as there are units in the index of the required power, ud the last product will be the power required. Example, Cube x±,z and 'Hx — ^z. X ±,z 2d? — 3j2r 2a? — S^r x±z z-*' r±xz ±xz^ QnaTes 3^±^XZ X ±z + Z^ .... x'±iix'z ± x^z + ^ Cobes \X^±%3?Z + Sxz* ■hz\ 4 a?' — Qxz — Qxz -\-9z' .4:x'''^l2xz + 9z'*- fix — 3z Sx^^Ux-z-^- 18;r;»* — 12«»;2f + 36a?;»« — 27;r^ The operation required by the preceding rules, however simple in ^ nature, becomes tedious when even a binomial is raised to a liigii power. In such cases it is usual to employ Sir Imoc Newton's Rule /or involving a Binomial. 1. Tojind the terms without the coefficients. — The index of the first, or leading quantity, begins with that of the given power, and de- cwisea continually by 1, in every term to the last; and in the following quantity the indices of the terms are 0, 1, 2, 3, 4, &c. 2. To find the uncice or coefficients. — The first is always 1, the •Kond is the index of the power : and, for the others, if the coeffi- cient of any term be multiplied by the index of its leading quantity, ttd the product be divided by the number of t^rms to that place, it ^giie the coefficient of the term next following*. ilToiff.— The whole number of terms will bo one more than the Wex of the given power; and when both terms of the root are + , ill the terms of the power will be + ; but if the second term of the foot he — , all the odd terms will be + , and the CTen terms — . * This role, expressed in general terms, is as follows : — (• + 4), - «• +».«"-' 6 + n. L=J a-«6« + n . !L±i.'LzJa"-«i«,&c 2 2 3 The lame theorem applied to fractional exponents, and with a slight modifi- ation, ssrres for the extraction of roots in infinite series ; as will be shown a ^&rtheron. 62 INVOLUTION. [part I. Eapamples. 1 . Let a '\- X he involved to the fifth power. The terms, without tho coefRcieDts, will he a\ a^ Xy d^x\ a^ x\ ax*y x\ and the coefficients will he ^ 5x4 10 X 3 10 X a 5 X 1 ' ^' 2 ' 3 ' 4 ' 5 ' and therefore the fifth power is a' + ba*x + lOa'x- -h lOa^a?' -f 5aa?' -f jr\ Here we have, for the sake of perspicuity, exhibited separately the manner of obtaining the several terms and their respective co- efficients. But in practice the separation of the two operations is inconvenient. The hest way to ohtain the coefficientB is to perform the division first, upon either the requisite coefficient or exponent (one or other of which may always be divided without a remainder), and to multiply the quotient into the other. Thus, the result may be obtained at once in a single line, nearly as rapidly as it can be written down. 2. {x ^ yy ^ x'' -\-l a^ z ^ 21 X' z' -\- 35 a?* z' + 35 x^z^ -f 21 x'z^ -f 7a?2^ -f z*. 3. {x ^ zf :=^ 3^ -^^x^ z -\- 28 :i^ ;f*^ - 56 x^ z' + 70 x' z' — 50 X' X?^ 4- 28 X' ;?« - 8 4? ;J^ -f ;^^ For Trinomials and Quadrinomiah. — Let two of the terms be regarded as one^ and the remaining term or terms as the other ; and proceed as above. Example. Involve X -\- y — z to the fourth power. Let X be regarded as one term of the binomial, and y — ^ as the other : then will U + ^ — zY = {;i? + ( j^ — -2?)}^ = a?* i- 4 ar* (y — z) + 6 a?*^ (^ — z^ -f A: X {y ^ zy + {y — z)\ where the powers of y — z being expanded by the same rule, and multiplied into their respective factors, we shall at length have a?* -f ^ x^y ^ Ax^ z •\- ^x'f — l^a^y z + 6aj»ar' + 4 xf — V2xy'z + ^'Hxyz' — 4 a? ^r* -h y* — 4 y** ^ + 6 / z^ — 4 jy 5r* -f- z\ the fourth power required. Had {jX -f y) and — z been taken for the two terms of the bino- mial, the result would have been the same. Note. — The rule for the involution of multinomials is too complex to be given in this place. CHiP. II.] KVOLUTION. 8.1 Sbct. VI. Evdutian. 1 . To find the roots of monomials, iSv/e.— Extract the correspondiDg root of the coefficient for the Mw coefficient: then multiply the index of the letter or letters hj the index of the root, (fractionally expressed, as explained at paee 4,) tk result will he the exponents of the letter or letters to he placed liter the coefficient for the root required. Examples. I Find the fourth root of 81 o*;?*. First \/ 81 = >/9 = 3, new coefficient. Then 4 X -J = 1, exponent of a; and 8 x J = 2, exponent of z. Hence 3 a :2^ is the root required. 2. To find the square root of a polynomial. Proceed as in the extraction of the square root, in arithmetic, as explained st page 29. Examples. 1. Extract the square root of a* + 4 a"* a? + 6 a^ ar^ -f 4 a a?* +ar'. a* + 4 a^x -f Qa^x"' -f Aax' + x* {a^ + ^ax + ara* tf« [root required. 4a' a? + 4a=;r^ •io' + aaa? 2 a if X- 2a' x^ + 4a;p» - x- 2. Extract the square root of :r* — S a;^ + * 4?* — i a? + ^^ , i X 1 ^ 2 ^ 16 a 1 ' 2 ^ 16 64 EVOLUTION. [part I. 3. To find the roots of powers in general. When the power (w) of the root to he extracted is not very high, the following method may he employed for extracting the nth root*. Rule. — Range the several terms in the order of the powers of one of the unknown quantities ; having done which, extract the nth root of the first or leading term, which place in the quotient, and cancel such term ; then bring down the second term for a dividend, which divide by the term of the root just found, raised to the power of (n — 1 ), and multiplied by n, the quotient will be the second term of the root. Then to find the third term of the root, involve the two already obtained to the nth power as far as the third term, which sub- tract from the third term of the given power for a dividend, which divided by the same divisor, gives the third term of the required root. And, in like manner, to find the fourth term of the root, involve those already obtained to the wth power as far as the fourth term, which subtracted from the fourth term of the given power, gives a fresh dividend ; and thus proceed until the root is extracted. Examples. 1. Extract the cube root of a?« — 6 ar' + 15 a?* — 20 «* + 15 ar* — 6a? -h 1. ^P**- 6a?^+ 15a?*-20af» + 15d?' — 6« + 1 (a?'— -2^?+ 1 «** [jroat rehired. 3a?* * -6ar^ irB-«6«^*-fl2a?* = (a;« — ^f to the third term. 3a?» j,*-_6*» -h 16**~20*"+l5*«-6*+l=(*>-2*+l)* 2. Find the 4th root of 16a* — 96 0=* a? + ^K^a^x — 216aj?' 4- 81 x\ 16a*— 96a^a?-f 216a-a?' — 216a;r'» f 81;i?»(2a-3^ 16 a* 8 a"* X 4= 32 a")*- 96 a^^ 16 g* — 96a-^a? +216o»;r' — 2l6o^-|-81 ** « (2a - 3*V * * * * * * By this mode of expreision is meant any root whatever; the nile may be made to apply to any particuhir case by substituting for n, the power of the root required, which in the first Example above is 3, and in the second is 4. CHIP. II.] 8UBD8. 05 ilTdtH.— In the higher roots proceed thus : — For the liqiMdraiey extract the tguare root of the square root. n nk4 root, ,, cube root of the square root. „ ei9M root^ „ sq. rt. of the sq. rt. of the sq. rt. n nmtk rooty ,, cube root o£ the cube root. Exunples, however, of snch high roots seldom occur in any prac- tical inquirieB. Sect. VII. Surds. A Swrdy or irrational quantity, is a quantity under a radical sign or fractioDal index, the root of which cannot he exactly obtained. (See Abith. Sect. 9. Evolution.) Sards, as well as other quantities, may be considered as either ample or compound, the first being monomials^ as v^S, a*, ^a b\ the ^n poly nomials, as y/3 + v'S, V« + >/*— v'c^* V(«— >/^)' ^a + 3, &c Radons] quantities may be expressed in theybrm of surds, and the operation, when effected, often diminishes subsequent labour. RBDUCTION. 1. To reduce eurds into their simplest expressions. Ca8B I. If the surd be not fractional, hut consist of integers or in- tegral factors under the radical sign : Arfe.— Divide the given power by the greatest power, having the wne mdex, contained therein, that measures it without remainder; let the quotient be affected by the radical sign, and have the root of tbe diiieor prefixed as a coefficient, or connected by the sign x . Examples, 1. v^75 = >/(26 X 3) = v'aS X >/3 = 6 ^S. a. \/US = X/(U X 7) = V®^ X V = ^ V^. 8. V176 = V(1«X 11) = V16 X V11 = ^V11- 4. v^(8af»— 12«*^) = >/4a*(2«-8^)=s >/4«* X v^(2a?— 3^) = 2« v'(2«— 3^). «. V(56^y + 8^) = VQ^'C^^ -f 1) = V^*^ X Cabb II. ^ M« nircf be fractional, it may be reduced to an equiva- ^mteqralone, thus: — . iMe.— Multiply the numerator of the fraction under the radical *|n bj that power of its denominator whose exponent is one less F 66 SUBD8. [PABT I. than the exponent of the surd, and place it as a whole number under the radical sign, and prefix to it for a coefficient, the coefficient of the original surd (whether unity or any quantity) divided by the deno- minator of the given fraction taken from under the radical sign. Note, — This reduction saves the labour of actually dividing by an approximated root ; and will often enable the student to value any surd expressions by means of a table of roots of integers. Ejeamples, 1. v^^; then I x 3 = 3, and | v'S = v'f a. v^|; then 1 x 6 = 5, and |V6 = y/^. V- : then 1 x n = n, and - ^n = v^ -. 71 n n VI; then 4 X 5« = 100, and ^ V^^^^ = Vf 3. 3 /2 a then 2 a X 5a?|* = 50 a«*; and -— VSOaar* I 6a! 6. VM = 2 VA» ^^en 2 X 81»= 13122, and ^\V13122 = |V18=VM- Case TIL If the denominator o/ the fraction be a binomial or r««f- dualy of which one or both terms are irrational and roots of%quare% : Rule.^ Multiply this fraction by another which shall have its nume- rator and denominator alike, and each to contain the same two quan- tities as the denominator of the given expression, but connected with a different sign. ■ Note 1 . — By means of this rule, since any fraction whose nume- rator and denominator are the same, is equal to unity ^ the quantity to be reduced assumes a new appearance without changing its value ; while the expression becomes freed from the surds in the denomma- tor, because the product of the sum and difference of two quantities is equal to the difference of their squares. Examples. 1 S _ 8 >/5 -f >/3 _ 8(v^5 H- >/3) n/5— >/3 >/5— v^3V5 + n/S"" 2 = 4(^/5 + v^3). 2. 3 3 V5- v^2 _3(v^5- y/a) v^5 -f >/2 v^5 + ^^2 V5 - ^^2 8 = >/5 - >/2. CHIP. 11.] 9U11M. 67 V10O-2V6O-I-V36 16-2^^60 ^ = 5I3 ^ = ^=8-^^60 = 8-2^15. 4 >/^^ :^ n/^i^ V5-t/3 ^ V5-V 3 V5-^V3 V^-^V3'V5-V3 n/5->/3^^ Abte2. — Upon the same general principle any binomial or re- «doal surd, as V^ — V^ ™*y ^® rendered rational by taking VA;-'t v(a— 'B) + V(A"~'BO T V(A""^BO + &c- for a Doltipfier : where the upper signs must be taken with the upper, the lower with the lower, and the series continued to n terms. Thus, the expression ^/o' — \/6', multiplied by \jc^ •\' \/ a h' + V^+ V**> gives the rational product a* — ft\ ^ To reduce Bwrds having different exponenU to equivalent ones that have a common exponent, Inrolre the powers reciprocally, according to each other s exponent, for Dew powers: and let the product of the exponents be the common exponent ^ofe. — Hence, rational quantities may be reduced to the form of ttj isaigned root; and roots with rational coefficients may he so Ktiooed as to be brought entirely under the radical sign. Examplee. 1. a"and 6", become a"'" or a"" and ft*"'" or A*"". 2. a* and 6% become a * ^ or a* and ft* * * or 5* . 3. si and 2*, become 3* and 2^, or V3' and \/^\ or %/^l 4. (a + ft)*, and (a - ft)*, become ^\/{a + ft)' and »V(« - *)'• 5. The rational quantity a', becomes >/a*, V^> ^/®^ ®' V^"*- 6. 4aV5ft, becomes V(^«)' X V^^ V^^o^x V^^ <»• These and other obvious reductions which will at once suggest tiiemselTes, being effected, the operations of addition, subtraction, &c. ire 80 easily performed tipon such surd quantities as usually occur, ^ it will suffice merely to present a few examples without <J«tailing rules. f2 68 SURD9. [PABT I. ADDITION. EofA. v'SH- v'18=-s/(4.2)+v'(9.2) = 2v^2 + 3^/2=6>/2. 2. Add together v^54, ^/^, andv^^. ^^64 = >/(9 . 6) = v'9 X >/6 = 3 v'e ] The sum of these is ^/* =n/A=>/Ax >/6 = *n/6 (3-hi + |)^/6 ^/A = n/(V* • i) = >/ « = -J n/6 J = S^lff ^6. 3. >/27a«ar+ v'Sa'ajrs v'COa* . 3«) + (a^ . 3«) = 3a»v'3« + a>/3a? = (3a« + a)N/3aT. SaVJ + fl" V* = (8« + «')V*• 8UBTRACTI0N. ^a?. 1. 2>/50- ^/18 = 2 ^/(25 . 2) - >/(9. 2) = 2 . 6 >/Q - 3 v^2 = (10 - 3) V2 = 7 V2. 4. V^S^^'^ - V10«* « = V(1^5a* . 2a?) - VC^a' • 2«) = 6aV2« — 2aV^* = SaV^a?. 5. >/46**«— v'20«^ar»= >/(9**. 6a?)— v'(4a«ar*. 6*) = (3^«2*a?)>/6a?. a» - c« / 1 \ac/ . MULTIPLICATION. ^a?. 1. \/lS X 5 V4 = 6 V(18 . 4) = 6 V(4 .2.9) = 6 »y (8 . 9) = 6 . 2 V^ = 10 V^- 2. |^/■^xiVA = f.i^/(i.^Jff) = i^/^^=i^/(^.^) = i>/:^(fty = i-AV36 = ^v'36. 3. a* X a^ = a^"*"i = a'A"*""^ = a+^, 4. (a? + z)^ X (a? -f ^F = (« + ;?)*"^* = (« + ^)^. 5. (a? + >/y) X (a? — Vy) = a?» — y. 6. (« + >/y)* X (ar - ^y)i =^{0?^ y)^. '^. jjr" X 3^ = «" *" = j2r "*" . CHAf, II.] SURDS. 69 10. (T X (f^ =€r X -. = flT— . a 11. V— ax >/ — asv'aV— 1 X v'a>/— l=ax— 1=— a. 12. n/— ax^/ — ft= v'a>/— 1 X v'5>/— l = >/<lft X — 1 DIVISION. iSp.l. VlOOO-5-aV^ = lVT =^V250 = 2V(125.2) = 10 V2. =«V(A-8) = H.fV8 = MV3. 8. «i ^ a?* = a?^* =: a?"^ = 1 ^ aj*. II II »— II 4. /^ «- = «•- = «— . 6. "^^^/(^*-«^)-^-^ ^(a-^) = ^^x INVOLUTION. &.I. (|a*)«=:|.|.a*+* = ia*=:iV^- 2- (J^/iy = ^i•i•^/(J•i•i) = :5V^/4 = lV^/(4•|) 8. (8 + ^/5)« = {(3 + V6) (3 + >/6)} = 14 + 6 ^/5. 4. (a- v'J)* = «'-3a» >/* + 3a6-6v'*. EVOLUTION. i&.l. v'lO' = v'lOOO = V(100 . 10) = VIOO X vio = 10^/10. 8. V31aV« = V(91<**y*-5^«) = ^«Wy'^• 3. ^(a* — 4 a >/5 + ^) = a — 3 ^/5, the operation being P^ormed as in the arithmetical extraction of the square root Aofe. — The $quare root of a binomial or residual a ± 5, or even ^ i tiinomial or quadrinomial, may oflten be conveniently ex- *neted thus:— Take d = V(a» — ft*); then >/(a ± ft) = V ^ ± V ^^' ^"^ "^ evident: for, if /y/^^ ± 70 SIlfPLS EQUATIONS. [PABT I. A / — — be squared, it will give a 4- >/(a* — <^) or a + d, as it ought : and, in like manner, the square of a / — a / —5 — * is a — >/(a^ — d^)i or a — 6. Ex. 1. Find the square root of 3 -j- 2 v'jJ. Here a = 3, ft = 2 V^, </ = ^(9 — 8) = 1, = Vi + N/f = >/2 + n/1 = 1 + >/2. 2. Find the square root of 6 -* 2 >/5. Here a = 6, ft = 2 >/5, rf == >/(86 - 20) = Vl« = ^ 3. Find the square root of 6 + >/8 — >/12 — >/24. Here a = 6 + ^/8, ft = >/ia -f n/24, rf = v^(6 + >/8)» — (n/12 + v^24)-^ = ^/(44 -f 12 ^^8 — 36 - 2 ^^12 . 24) =: >/(44 - 36 + 12 n/8 -- 12 -s/8) = v^8. o + rf 6 + 2 ^^8 Q.,o A^ — ^ Conseq. -y- = ^-^^ = 3 + >/8, and -^ = 6 + >/8-v/8 . 2 =^- But (Ex. 1), v^(3 -f 2 >/2) =^ ^^(3 -f ^/8) = I -f >/2. Therefore the root required is 1 + >/2 — >/8*. Sect. VIII. Simple Equations. An algebraic equation is an expression by which two quantities, called members (whether simple or compound), are indicated to be equal to each other, by means of the sign of equality (=) placed be- tween them. In equations consisting of known and unknown quantities, when the unknown quantity is expressed by a simple power, as Xy x\ x\ &c, they are called simple equations^ generally ; and particularly, ample or pure quadratics^ cubics^ &c. according to the exponent of the nn- * For the cube and higher rooto of binomials, &c. the reader may oontult the treatises on Algebra, by Madaurin, Emerson, Lacroix, Bonnyoastle, J. R. Young, and Bine. CliP.U.] 8IMPLI EQUATIONS. 71 known quantity. But when the unknown quantity appears in two or more different powers in the same equation, it is named an ad^ /eetorf equation. Thus a;*- = a + 15, is a simple quadratic equation : 2^ f a f = 6, an adfected quadratic. It is the former class of equations that we shall first consider. The reduction of an equation consists in so managing its terms, tint, at the end of the process, the unknown quantity may stand •lone) and in its first power, on one side of the sign =, with the ham quantities, whether denoted by letters or figures, on the other. Thus, what was previously unknown is now affirmed to be equ4d to the aggregate of the terms in the second number of the equation. ^In general, the unknown quantity is disengaged from the known wet, hfpefforming upon both members the revbbsb opbbations,"* to those indicated by the equation, whateyer they may be. Thus, If any known quantity be found added to the unknown quantity, jetitbesabtracted from both members or sides of the equation; or if any sacb Quantity be found subtracted, let it be addedf . If the nuKnown quantity have a multiplier, let the equation be dJTided by it. If it be found divided by any quantity, let that become the mul- tiplier. If any power of the unknown quantity be given, take the corre- ipondiug root ; or if any root, find the corresponding power. If the unknown quantity be found in the terms of a proportion [ir^ Sect. 10), let the respective products of the means and ex- ^es constitute an equation ; and then apply the general principle, ts above. Examples. 1- Given x — 3 + 5=9, to find x. Otherwise, in appearance only, not in effect. By transposing the 3, and changing its sign, d; + 5 = 9 + 3. By transposng the 5, and changing its sign, d; = 9 + 9 — *=: 7. ^ Given 3 « + 5 = 20, to find x. First, by transposing the 5, 3 j; =x 20 — 5 «= 15. by dividing by 3, j; =s y ss 5. * This timple directioii, comprehending the seven or eight particular rules ^the rednction oi equations given by most writers on algebra, from the time *f Ncvtoo down to the present day, is due to Dr. Hutton. It is obviously ^Knded npon the mathematical axiom, that equal operations performed upon ifi*! things produce equal results. t These two operations constitute what is usually denominated trantpotUum^ ^cenae the operation of thus adding or subtracting any quantity from each side *f the eqoatiim is moat simply perwnned by tron^poiinf it from one side of the ■fisiion to Che other, and dianging iu sign. 72 SIMPLE EQUATIONS. [PAI 3. Given - -f </= 3 6 — 2<:, to findd?. First, transposing dy - = 36 — 2c + </. a Then, multiplying by a, xs=3ab^^ae-{-ad, 4. Given V(3* + 4) -f 2 = 6, to find a. First, transposing the 2, 51/(^* + ^) = ® ~ ^ = ^• Then, cubing, 3« + 4 = 4^* = 64. Then, transposing the 4, 3 « = 64 — 4 = 60. Lastly, dividing by 3, a? = ^o = 20. 5. Given 4a^— 56 = 3£^;p + 4tf, to find a. First, transposing 5 b and S da^ 4aa; — 3(f;p = 56 + Then, by collecting the coefficients, (4 a — Sd)a!^bb -^ .'. by dividing by 4 a — 3 </, a? = r r-y 4 a — ott 6. Given |a; + j^^ — j^^=3, to find a. ^f^YS ^ ^*^2 = I 80* + 24x - aO« = 360. 4 X 6 X 6, we have ) ^ That is, collecting the coefficients, 34 a; = 360. .-. dividing by 34, a? = ^ = i^ = 10 -J^. 7. Given ^x : a : : 6 6 : 3 e, to find x. Mult means and extremes, ^ c a; s: 5 a 6, T%. .,. I o ..1 « 20 a 6 Dividing byfc, a;=:5ao-ri<? = -^ • 8. Given a '\- a =^ >/d^ 4- « >/(4 6- + a?'), to find a?. First, by squaring, we have, a' + 2 a a? + «' = < a? v^(4 6* + a?^) Then, striking out a* from both sides, 2 a a? -f- ^ X >/(4 6» -i-a:^) dividing by a?, 2 a + a? = >/ (4 6* 4- a?*) squaring, 4 a' + 4 a a? + a:** = 4 6* + a?^ striking out a^ and ) ^, _ , transposing 4a%) A' M- K . 46^-4a« 6' dividing by 4 a, a: = = a. 4 a a 9. Given \/ca? — ac = 6 H- y^* — «> to find a?. First, dividing by v'^ — «> we have -^/c = —z — — — ^. CHi?. n.] BXTBBMINATION. 73 V'C— I h tranqK>8iiigthe 1, ^/c — I, or 1 ^{x - a) mTerting and transpodDg the fractions, — ^—7 -"^sr moltiplying by d, >/(« — a) = v'c — 1 sqaanng both sides, 4? — a = — ; — —r- iln^.^: 10. GiTen 18 — >/3 « = >/l3 -f 8 ar, to find a?. An$. X = 12. 8 11. GiTen y + ^/ 4 + y* = 7/4 . ^8-) > ^ ^°^ y* 12. GWen J (* + 1) + i (* + ^) == H« + ^) + 10> ^ fi°<l*- il9M. d? = 41. 13. Giyen^^A/^^ s/(a:— 1) :; 3 : 1, to find a:. AnB,\\. 14 Given (6* + «*)» = (a» + ^)*, to find «. EZTBBMINATION. Wlen two or more unknown quantities occur in the consideration of ID algebraical problem, they are determinable by a series of given ixlepeDdent equations. In order, however, that specific and finite Miotioos may be obtained, this condition must be observed, that there ^gwm as many independent equations as there are unknown quan- ^. For, if tibe number of independent equations be fewer than tlie unknown quantities, the question proposed will be susceptible of ID indefinite number of solutions'!' : while, on the other hand, a greater number of independent equations than of unknown quan- tities, indicates the impossibility or the absurdity of the thing at- ten^. Where two unknown quantities are to be determined from two in- ^^dent equations, one or other of the following rules may be em- ^vaknown quantities by two equations ; and so on. Thus, if J? + %/y = « -i- >/b and a — ^v = c — ^d Then #«a,y»6, sr = o, «=«rf. 74 EXTBRMINATION. [PABT I, Bide 1 . — Find the value of oDe of the uDknown letters in each of the given equations ; make those two values equal to one another in a third equation, and from thence deduce the value of the other un- known letter. This substituted for it in either of the former equa- tions, will lead to the determination of the first unknown quantity. 2. Find the value of either of the unknown quantities in one of the equations, and substitute this value for it in the other equation : 80 will the other unknown quantity become known, and then the first, as before. 3. Or, after due reduction when requisite, multiply the first equa- tion by the coefficient of one of the unknown quantities in the second equation, and the second equation by the coefficient of the same un- known quantity in the first equation : then the addition or subtrac- tion of the resulting equations (according as the signs of the unknown quantity whose coefficients are now made equal, are unlike or like) will exterminate that unknown quantity, and lead to the determina- tion of the other by former rules. Notes. — The third rule is usually the most commodious and expe- ditious in practice. The same precepts may be applied, mutatis mutandis^ to equations comprising three, four, or more unknown quantities : and they often serve to depress equations, or reduce them from a higher to a lower degree. Examples. 1. Given d^jf* -h 3y = 41, and Sar* — 4y = 12, to find x and y. 1st equa. x by 3, gives lHaP -f 9y = 123 2ndequa. x by 4, gives 12 j?* — 16 y = 48. The difference of these, 25 y = 75, whence y = 3. Then, from 2nd equa., 3 jr» = 12 + 4y = 12 -j- 12 = 24 Whence dividing by 3, ar* = 8, or a? =: 2. Ex. 2. Given a? -h y -r xr = 53, ;i? -f 2y + 3;8f = 105, and « + 3y + 4;2r= 134. 1. X -{- y -h J2r= 53 2. 0? 4- 2y 4- 34r=: 106 3. 0? -f 3y + 4;t= 184 4. 1st equa. taken from 2nd, gives y -f 2« = 52 5. 2nd equa. taken from 8rd, y -h jp = 29 6. 5th equa. taken from 4th, jt ss 28 7. 6th equa. taken from 5th, y = 6 8. 5th equa. taken from 1st, x = 24. Ex, 3. Given x-^yssa, x-^ zssb^ y -\~z^Cy to find Xy y, and s. 1. ar -f-y = a 2. X -^z= b 3. 1/ -^ z^si c (IIP. II.] OSNUUL PROBLEMS. 75 4. lft + 2nd + 3rd, gives 2ar + 2y + '^^^==0+ i + c. 5. Ha]f 4th equa. gives ^-ry + ^ = ia + 4^ + i^- 6. 3rd equa. taken from 5th, gives ^=|a-f|5— Ic. 7. dud eqna. taken from 6tfa, ^ = 1^ — 1^ + 1 c 8. 1st eqoa. taken from 6tfa, Jtrss _|a + i^ -\- ic. Ex, 4. Given ax -\-by^ e^ and o^^p -f ^'y as ^, to find 47 and y. cV — h<f , atf — ca^ &.6. Given ajp -f ^y + ^Jjzrassrf, c^x + V y -rf^ z ^ df of' x + 5"^ + c^'ar =: rf" to find a?, y, and ;2r. ^""a^V - ac'6" -f CO' ft'' - 6aV' + hifa^' - c5V '"aftV' — ac'ft" +ca'ft"-fta'c" + ftc'a"-cA'a''' & 6. Given d:(;p -h y + ^) = 18] y (^ + y -r ^) = 27 I to find a?, y, and z. ^(^-f^ + 4 = 86 J iln«. :i?=:2, y = 8, j2raB4. ^.7. Given (a? + y)- « 60, and(ay+y)- = 2^, to finda?andy. y ^ Ans. ^ = 10, y = 2. ^. 8. Given 4* + 4y + i;? = 62] l^+|yH-i^ = 47[ to find a?, y, and 4?. ilw*. a? = 24, y = 60, ;2r = 120. SOLUTION OP GENERAL PROBLEKS. A general algebraic problem is that in which all the qaantities ^^Bconed, both known and unknown, are expressed by letters, or ^general characters. Not only such problems as have their ttnditions proposed in general terms are here implied ; but every ^^f^cdar numeral problem may be made general^ by substituting ^'^ for the known quantities concerned in it ; when this is done, ^ problem which was originally proposed in a particular form ^^^^taes general. In solving a problem algebraically, some letter of the alphabet must w mbstituted for an unknown quantity. And if there be more QBbown quantities than one, the second, third, &c., must either be ^^piwied by means of their dependence upon the first and one or <W9 of the data conjointly, or by so many distinct letters. Thus, ^>Hay separate equations will be obtained, the resolution of which. 76 OBNBRAL PB0BLBM8. [PABT I, by some of the foregoing rules, will lead to the determination of the quantities required. ExampU$. 1. Given the sum of two magnitudes, and the difference of their squares, to find those magnitudes separately. Let the given sum be denoted by «, the difference of the squares by d; and let the two magnitudes be represented by x and y respectively. Then, Uie first condition of the problem expressed algebraically is « + y = *. And the second is oj' — ^ = d. Equa. 2 divided by equa. I, gives x^y=i- Equa. 1 added to equa. 8, gives 2 a; = - + « = Equa. 4 divided by 2, gives x = Equa. 5 taken from equa. 1, gives y^zs^ 9 8 %8 «* + D «•— D ^8 ^8 To apply this general solution to a particular example, suppose the sum to be 6, and the difference of the squares 12. Then » = 6 and D= 12, «« + D 86 + 12 48 , and X = — - — = — -- — = — = 4. 2« 12 12 «* — D 86-12 24 ^ and y = — r^ — = — rr — = —- = 2. ^ 2« 12 12 Suppose, again, 8 = 6, d = 5 : . 25+6 ^ . 26-6 ^ then X = — Jq— = 8, and y = — Jq" "^ ^ Ex. 2. Given the product of two numbers, and their quotient, to find the numbers. Let the given product be represented by/?, the quotient by ^; and the required numbers by x and y, as before. Then we have, 1. xy^=^p^ X and 2. - = ^. V Equa. 2 X by y, gives, xssqy Substituting this value) ^^ ^^ of X for it in equa. 1 J ^^ ^ P Dividing by ^, y* = — Extracting the square root, y = a/^ CHI?. 11.] QUADRATICS. 77 Then, \j snbstitation, a? = yy = ^ ^ /?. = ^ /?^ = ^//^y• Suppose the prodnct were 50 and the qaotient 2. Theny=^^=^^= V26 = 5, and «= V/?y = ^/lOO = 10. Agun, rappoee the product 36, and the qaotient 2| . Then y = y^| = ^^ =^/16 = 4,anda^=V;>y=: V81 = 9. Ex. 3. OiTen the sum (<) of two numbers, and the sum of their aqoues s, to find those numbers. iw. « = J * + J y^ 2 8 — ««, and^ = J « — J v^2 s — **. £x. 4. The sum and product of two numbers are equal, and if to eitber snm or product the sum of the squares be added, the result inll be 13. What are the numbers l—Ans. each = 2. Ex.5, The square of the greater of two numbers multiplied into & leaB, produces 75 ; and the square of the less multiplied into the prater produces 45. What are the numbers ? Bx, 6. A man has six sons whose successive ages differ by four Jem, and the eldest is thrice as old as the youngest. Required their •emalages?— iliw. 10, 14, 18, 23, 26, and 80 years. Sect. IX. Quadratic Equaiionx. QuiDRATic Equations are such as contain the square of the unknown ^UDtitj, and which, after due redaction, may be made to assume the Seseral form Adr' + B« + <' = 0; then dividing by a, the coefficient ^the first term, there results «^ -|- - « H- - = 0, or, making pss- ^ A A 9 = -, we have a!» + jD« + y = (1) *n equation which may represent all those of the second degree, J^Vidg being known numbers, either positive or ne^tive. Ut a be a number or quantity which, when substituted for ^, rcn- fe8^+/?a?-f ^ = 0; thenar ^ pa -f ^ = 0, or y== -^a^^pa. ^Wqaently j?' + jt> « + y, is the same thing as;i^ — a*+J^^~" ^«, or as (« + a) (« — a) -^ p (a ■— a), or, lastly, as (a? — a) («+a + ;?). The inqairy, then, is reduced to this, viz. to find all the values of * which shall render the product of the above two factors equal to *^g. This will evidently be the case when either of the factors 78 QUADRATIdS. [?ART I. is = ; but in no other case. Hence, we have x-^Oy ^0, and « -h a -f J^ = 0, or ;i? s=s a, and a? s= — a — p*. And hence we may conclude — 1. That every equation of the second degree whose conditions are satisfied by one value (a) of a?, admits also of another value ( — a — p). These values are called the roots of the quadratic equation. 2. The sum of the two roots a and — a — p is =^ — p; their pro- duct is — a^ — apy which as appears above is = g. So that the eoefi- cienty jD, of the second term is the sum of the roots with a contrary sign ; and the known term^ q^ is their product, 3. It is easy to constitute a quadratic equation whose roots shall be any given quantities b and d. It is evidently a^ — (6 -|- rf) X -k-hdzzz 0. 4. The determination of the roots of the proposed equation (1) is equivalent to the finding two numbers whose sum is — jp, and pro- duct q. 5. If the roots b and d are equal, then the factors x — b and x^d are equal ; and ar •\' p x -^ q \s the square of one of them. To solve a quadratic equation of the form a? •\' px ^ ^ =: 0, let it be considered that the square of ^ + | j9 is a trinomial, a^ + /> a? + \jfy of which the first two terms agree with the first two terms of the given equation, or with the first member of that equation when q is transposed. That is, with aj^ + jt? a? = — y. Let then \f^he added, we have 0^ -{- px -^-^p^ ^ \p^ — q of which the first member is a complete square. Its root is a? -f ^jt? = ± >/ {\p^ — q) and consequently iP = — i j» ± 'J{\j^ — q) otherwise, from number 2 above, we have X -\- of := — p and xaf =^q. Taking 4 times the second of these equations from the square of the first, there remains a^ ^^xaf + ^=jp* — 4j' Whence, by taking the root, a? — a?' = >/(/?* — 4 y) Half this added to half equa. 1, gives And the same taken from half equa. 1, gives which two values of a evidently agree with the preceding. It would be easy to analyze the several cases which may arise, ac-^ cording to the different signs and different values, of/? and q. Bat * If it be affirmed that the given equation admitN of another valae of jt^ besides the above, h for instance, it may be proved as before that # — 6 mnst be of the number of the factors of «■ + p * + 9, or of (* — a) (* + fl + ji). But » — a and jr -f a -f p being prime to each other, or having no oommon factor, their product cannot have any other factor than they. Conieiiuently 6 must either be equal toaorto — a— p; and the number of roots is restricted to two. CHIP. II.] QUADRATICS. 79 tliese need not here be traced. It is evident that whether there he flTCB 1. 2. 3. 4. ^ + />* = q 3^ ^ px == q 3^ + px = — 3^ -- px = — 9 The general method of solution is by completing the square^ that ii, adding the square of |/>, to both members of the equation, and tlien extracting the root. It may fariber be obsenred that all equations in which there are two tenns inTolving the unknown quantity or any ftinction of it, and tbe index of one double that of the other, may be solved as qua- dntics, by completing the square. Thus : It M ^ + pi» = ^, «*■ i />«* = y, a?^ ± /> a?^ = y, {jf^ -^ p X -\- qf ±(^ -{-px ^ q) =zr,{af^ -^ ary ± (aj*" — j^) = y, &c., are of tbe same form as quadratics, and admit of a like determination of die unknown quantity. Many equations, also, in which more than one unknown quantity are invoWed, may be reduced to lower dimen- Boni by completing the square and reducing ; such, for example, as (!? + /)» ± /» (:2r^ + y) = ^, 3 ± ^ = ^, and so on. Sr y N(fte, — In some cases a quadratic equation may be conveniently nhed without dividing by the coefficient of the square, and thus vithoQt introducing fractions. To solve the general equation a o^ ± *«=<;, for example, multiply the whole by 4 a, whence 4 a* a?® di iahx = 4 a <;, adding ^ to complete the square, 4ta^aP ± iab x + ft' = 4 a'c -f P taking the square root, 2 a « ± J = di *J(iae + ^); whence x = ^ — ^ }" -: which will Krre for a genera] theorem. Examples, I Giren x^ — 8 x + 10 = 19, to find x, transnosing the 10, dj» — 8a? = 19 — 10 = 9 completing the square, a?^ — 8« + 16s= 9-fl6 = 26 extracting the root, a? — 4 = ± 6 consequently a? = 4±:5 = 9or — 1. ^ Given -^ — ^ -— , to find the values of x, X ST 9 multiplying by «*, 10 a? — 14 + 2 a? = — — , tnnsposing, ^ «* — 12a? = — 14, dividing by V, ««-^«=-^, cmplet. squ. x» - 44 oi + (^^ = iH - « = -i¥r> extract, root, « — 4i » ± -fr, tmnspottng, x^^±^^S or ^. 80 QUADRATICS. [PARI I. 3. Given «*-f2a? + 4 >/«* -h 2 a? 4- 1 = 44, to find ». adding 1, we have (j;* -|- 2a: + 1) + 4 ^{x^ + 24? 4- 1) =45 complet squ. (a:* -f 2a! + 1) 4- 4 >/(jj* + 2a? -f + ^ = 49 extract, root, ^{^ -|-2a?H-l)-H2 = ±7 transposing the 2, ^{o^ -f 2a? +1) = ±7 — 2 = 5 or — 9 that is,a?4-l = 5 or — 9 hence a? = 4 or — 10. 4. Given a?" — 2 aa?^ = c, to find x. n complet. squ. a;" — 2aa?^ -|-a?=:c-fo^ extract, root, a:^ — a = ±: >/ (c + a*) n transposing, a?^ = a di ^^^ (c + o^) consequently, x =^ {a '±. y/c -f a*)". •E^ 4;r 6. Given ^4- — = 12, and a? — y = 2, to find a? and y. tr y 3^ X complet. squ. in equa. 1, — -f4-4-4=16. Extracting root - -f 2 = ±: 4 : whence - = 2 or — 6, and a?= 2y or — 6y. Suhstituting the former value of x in the 2nd equa., it becomes 2^ — y = 2, or y = 2 ; whence a? = 4. Again, substituting the 2nd value of a;, in equa. 2, it becomes — 6y — y or — 7y = 2; whence y = — ^, and a? = + V . 6. Given a?*y' — 5 = 4 a? ^, and i a? y = ^ y*, to find a? and y, equa. 1, by transposition, becomes a?*y* — 4a?y = 6 completing the square, a^}^ — 4a?y + 4 = 9 extracting the root, a?y — 2 == ± 3 whence a?y = 5 or — 1. Substituting the first of these values for a?j^ in equa. 2, it becomes |. y = ^ : whence ^ = 1 and a? = 5. Substituting the 2nd value in the same equation, it becomes I y = — J : whence y = — \j\ = — i V^^> *°^ a? = — 1 ilfw. a? = ± 3, or ± >/^. 8. A man travelled 105 miles at a uniform rate, and then found that if he had not travelled so fast by two miles an hour, he would have been six hours longer in performing the same journey. How many miles did he travel per hour ? Am. 7 miles per hour. CHAP. II.] EQUi^TTONS. 81 9. Find two socb numbers ttmt the sum, product, and difference of their squares may be equal. Ans. \ + i >/5, and ^ + 5 >/5. 10. A waterman wbo can row eleven miles an hour with tlie tide, ud two miles an hour against it, rows five miles up a river and back sgaJD in three hours : now, supposing the tide to run uniformly the same way during these tbree hoars, it is required to find its velocity ? Ans. 4J^ miles per hour. Sect. X. Equations in General. Equahons in general may be prepared or constituted by the maltipiication of factors, as we have shown in quadratics. Thus, snppose the values of the unknown quantity x in any equation were to be expressed by a, b, c, </, &c., that is, let a; = a, a? = 6, ^ = c, -p=(i, &c., disjunctively, then will x — a=:0, ar — ft = 0, ar — <?=0, *-rf=0, &c., be the simple radical equations of which those of tiie higher orders are composed. Then, as the product of any two of th^ gives a gvadratic equation, so the product of any three of them, as (4? — a) (4J — b) {x — c) =: 0, will give a cubic equation, or one of three dimensions. And the product of four of them will coosdtute a biquadratic equation^ or one of four dimensions ; and so OD. Therefore, in general, the highest dimension of the unknown fiOHHtff X is equal to the number of simple equations that are mul- fiplied together to produce it. When any equation equivalent to this biquadratic (x — a) (jx — b) (* t) {x — d) = is proposed to be resolved, the whole difficulty wniists in finding the simple equations a? — a = 0, x — 6 = 0, Jf-c=0, X — ^ = 0, by whose multiplication it is produced; for each of these simple equations gives one of the values of x^ and one wlotion of the proposed equation. For, if any of the values of x <ledoccd from those simple equations be substituted in the proposed equation, in place of x^ then all the terms of that equation will ^iih, and the whole be found equal to nothing. Because when it i« sopposed that ^ =: a, or x^b^ or ;p = c, or x z= dy then the product (4? — a) (x — b) (a? — c) (a? — d) vanishes, because one of the factors is equal to nothing. There arc therefore four supposi- tions that give (x — a) {x — b) {x — c) {x — <^) = 0, according to the proposed equation ; that is, there are four roots of the pro- posed equation. And after the same manner any other eauation •dmita of as many solutions as there are simple equations multiplied ^ one another that produce it, or as many as there are units in the behest exponent of the unknown quantity in the proposed equation. Bnt as there are no other quantities whatsoever besides these four K ^ c, d^) that, substituted in the proposed product in the place of ^ wfll make that product vanish ; therefore, the equation {x — a) (*-h){x — #r) (;p — df) = 0, cannot possibly have more than these 82 EQUATIONS. [part four roots, and cannot admit of more solutions than four. If ^ substitute in that product a quantity neither equal to a, nor 6, nor nor dy which suppose e, then since neither e — a, c— i, e — c, n e — dyis equal to nothing; their product cannot be equal to nothir but must be some real product : and, therefore, there is no suppo tion beside one of the aforesaid four, that gives a just value of according to the proposed equation. So that it can have no mc than these four roots. And after the same manner it appears, i\ no equation can have more roots than it contains dimensions of i unknown quantity. To make all this still plainer by an example, in numbers, suppc the equation to be resolved to be x^ — 10 ar* + 35 ;ir — 50 j? + 24 = and that we discover that this equation is the same with the prodi of (a? — 1) (j?— Q) {x — 3) {x— 4), then we certainly infer tl the four values of a: are 1, 2, 3, 4 ; seeing any of these numbe placed for Xy makes that product, and consequently a?' — 10 dr* 35 x^ — 50ii? + 24, equal to nothing, according to the propos equation. And it is certain that there can be no other values of besides these four : for when we substitute any other number for in those factors ii? — 1, d? — 2, x — 3, ;r — 4, none of them vanL and therefore their product cannot be equal to nothing, accoi'ding the equation. A vai'iety of rules, some of them very ingenious, for the soluti of equations, may be found in the best writers on Algebra*; 1 we shall simply exhibit the easy rule of Trial-and-Error, as it is giv by Dr. Hutton^ in the 1st vol. of his ** Course of Mathematics." Rule for the general solution of Equations hy Trial-and-Error, " 1 . Find, by trial, two numbers, as near the true root as possib and substitute them in the given equation instead of the unknot quantity ; marking the errors which arise from each of them. " 2. Multiply the difference of the two numbers, found by tri by the least error, and divide the product by the difference of t errors, when they are alike, but by their sum when they are nnlil Or say, as the difference or sum of the errors is to the differei of the two numbers, so is the least error to the* correction of supposed number. '^ 3. Add the quotient, last found, to the number belonging to t least error, when that number is too little, but subtract it when t great, and the result will give the true root nearly, " 4. Take this root and the nearest of the two former, or a other that may be found nearer ; and, by proceeding in like mann a root will be had still nearer than before ; and so on to any degi of exactness required. " Note — It is best to employ always two assumed numbers tl shall differ from each other only by unity in the last figure on I right; because then, the difference, or multiplier, is 1." * i'^ee the trentiseii of I/acroix, BonnycMtle, Wriod, J. R. Vming, &c. CBIMI.] BQUATIONS. 83 Example, To find the root of the eohic equation ^r* + ^v' + x = 1 00, or the kloe of X in it. Here it is soon found that x L^ between 4 and 5. Assume, b^refore, these two numbers, M^ the operation will be as fol- iSup. le 64 84 -16 2nd Sup. X . sums . errors 5 25 125 155 -^55 the som of which is 71. TbcnasTl : 1 :: 16 : -225. Hence x = 4*225 nearly. Again, suppose 4*2 and 4*3, and repeat the work as fol- lows : \9tSup. 4-2 17-64 74088 95-928 —4072 X x' x' 2nd Sup. 4-3 18-49 79-507 sums . . 102-297 errors . . +2* 297 the sum of which is 6*369. As 6-369 : 1 :: 2297 : 0036 This taken from . . 4*300 leaves x nearly = 4*264 Agiio, soppose 4-264 and 4*265, and work as follows : 4-264 18'181696 7:'526752 W-972448 -0-027552 X x^ x" sums 4-265 18-190225 77-581310 100036535 + 0-036535 the sum of which is •064087. Then as 064087 : -001 : : -027552 : 0-0004299 To this adding ... 4*264 gives X very nearly = 4-2644299 When one of the roots of an equation has been thus found, then ^t for a dividend the given equation with the known term trans- ited to the unknown side, so as to make the equation equal to •olkiog ; and for a divisor take x minus the root just determined : (he quotient will be equal to nothing, and will be a new equation o 2 84 PR0OBE8SION. [PABT depressed a degree lower than the former. From this a new Tat of X may be found : and so on, till the equation is reduced to a q%: dratic, of which the roots may be found by the proper rules. Example, Given the biauadroHc equation, «* — 1 1 «^ -f 28 «^ -f 36 ^e 144 = 0, to find the four roots. First, by the above method of Trial-and-Error, we find one oF roots to be 6 ; then, a!-.6)a?«— lla?'-f 28a?^ + 36a; — U4(a?^ - 5 «' - 2*4- a?«— 6 a?' — 6a?' -f-28a?' + 30a?^ + -.2 a?'' -2 a?* 36 a? 12 a? 24 a? — 144 24 a?— 144 The quotient of which, ar' — 6 a?*' — 2 a? -f 24 = 0, is a eMc ^^ tion of which one of the roots is found to be 4 ; then, -r - 4 )ar» - 6 a?* — 2 a? + 24 (a?^ — a? — ar»-4a?'* a?' — 2a? a?' + 4a? — 6a? + 24 — 6a? + 24 The quotient of which, a?^ — a? — 6 = 0, is a quadratic equation, which the two roots are readily found to be 3, and — 2 ; thus t J original biquadratic equation is composed of the factors (a? — (a? - 4), (a? - 3), and (a? + 2). Or, (a? + 2).(a?-3).(a?-4).(a?-6) = a?»--l]a?'»+28a^+36a?~144=t Sect. XI. Progression. When a series of terms proceed according to an assignable orde either from less to greater or from greater to less, by continual eqn <* CSIP. II.] ARITHMETICAL PB0GRBS8I0N. So diflerences or by saccessive equal products or quotients, they are said to fonn a/iro^retttois. If tbe quantities proceed by successive equi-differences they are «id to be in Arithmetical Progression, But if they proceed in the «iDe continued proportion, or by equal multiplications or divisions, ibev are said to be in Geometrical Progression. If the terms of a progression successively increase, it is called an ^Kotiing progression : if they successively decrease, it is called a ^ktcmding progression. Thus, 1, 3, 5, 7, 9, &c. form an ascending arithmetical * "^ ^2^ 20, 18, 1 6, &c. form a descending arithmetical > Progrcssi iion. 1, 3, 9, 27, 81, &c. form an ascending geometrical ^ ^ 1) !» 49 ^^' ^'^1'°^ A descending geometrical / ABITHMBTICAL PBOOBESSION. 1 Let a be the first term of an arithmetical progression, d the common difference of the terms, z the last term, n the number of terms, 9 the sum of all the terms. Tbcnei,a-f(i; a-^-'Stdj a-^-^d, &c., is an ascending progres-sion, and a, a^dj a — Sd!, a^^dj &c., a descending progression. Hence, in an ascending progression, a 4- (« — 1) <^ ^^ the last term ; m a descending progression, a — (n — 1 ) </, is the last term. ^Lcta series be a+{a + d) -f (a +2 </) + (« + 3 rf). TlKwne inverted (a + 3rf) -f (« + 2</) + (« + <if) + « ^«amofthetwo(2a-i-3rf)+(2a-f3<ir)4-(2a+8<Q-f(2a+8rf)=2*. Thitii, (2a + 3</) x 4, in this case (a -f a + 3rf)n = 2#. . Cowe^ntly, »=J(« + a"+"37)n, or = i(<» + ^)»»> "°<^ * >kie = a -f 3</. The same would be obtained, if the progression *^ descending; and let the number of terms be what it may. 3. From the equations ;? = a -f (« — 1) <^ * = 4 »* (<• + ^)» ^^^ '=*}«{a -^0 + (n-- !)<'}» ^6 ™*y readily deduce the following !^^<Kemi applicable to ascending series. When the series is descend- ^ other U)e signs of the terms affected with d must be changed, I g I *<BiQ8t be taken for z; and vice versd. « 2« /9\j * — « «* — fl^ ^zn — ^s 2# — 2na »— 1 2« — ;» — a n» — n n» — » (3.)* = a + nrf-d=* +i*rf- Jrf = 86 OKOMETRICAL PROGRESSION. [pART I. (4.) «=|w(a + ;2r) = (a-f ^«(/-|rf)n=:(z-^«</ + ^d) n " Ud (5.) w = = ; I. ^ ^ a -^ z d Examples. 1. Required the sum of 20 terms of the progression 1, 3, 5, 7, 9, &c. Here a =: 1 , </ = *^, w = 20 ; which heing suhstitutcd in the theorem « = (a + ^7«(/ — | </) 7«, it hecomes « = (1 -f 20— 1) '20 = 20 X Hi) = 400, the sum required. Note, — In any other case the sum of a series of odd numbers beginning with unity, would be = w^, the square of the number of terms. 2. The first term of an arithmetical progression is 5, the last term 41, the sum 299 Required the number of the terms, and the common difference. ^« 59H ,^ , , ^ Here n = = — - =: 13, the number of tenns, a -I- 5r 46 and d = = — -- — = 3, the common difference. w — 1 12 3. There are 8 equidifferent numbers : the least is 4, the greatest 82. What are the numbers ? TT * z -^ a 32 — 4 ^ , ,. _ Here d = 7 = — z — = 4, the common difference. n — 1 i Whence 4, 8, 12, 16, 20, 24, 28, 32, are the numbers. 4. The first term of an arithmetical progression is 3, the number of terms 50, the sum of the progression 2600. Required the last term and the common difference. Here ;2r = — — o = — ; 3 = 104 — 3 = 101, the last term, n 50 , ^ ;2r — a 101 — 3 ^ . and d = = — = 2, the common difference. w — 1 49 5. The sum of six numbers in arithmetical progression is 48 ; and if the common difference d be multiplied into the less extreme, the product equals the number of terms. — Required those terms. Ans. 3, 5, 7, 9, 11, and 13. GEOMETRICAL PROGRESSION. Let a be the first term of a geometrical series ; r the common ratio ; z the last term ; n the number of terms ; s the sum of all the terms. CHAP. II.] GEOMETRICAL PROGRESSION. «: . am: tea Then a, ra» r'o, r* a, r"""'a, is a geometrical progression, whicli will be tucending or descending^ according as r is an integer or zfraaioH, Let the prog, a -f ra -f r*a + r^a -f r*a = «, be x by r, it becomes ra -^ f^a -{- r^ a -{- r^a •\- r^a ^:^rs. The diff. of these is, — a -^^ r^a^^ rs — ». But r^a is the last term of the original progression multiplied by r, or in general terms r"""' a x r^ that is r" a. Consequently r*<i — a = r* — ». «T, r"a — a r" — I , ^ , whence « = = —a ^ the sum of the series. r — 1 r — 1 A amikr method will lead to a like expression for «, whatever be the nlue of fi. If r be a fraction, the expression becomes trans- lormed to # = a. 1 — r Now from these values of z and s the following theorems may be deduced. ; * -f r;2r — r*. r r" — 1 w*= j(r» — 1) _ q(l — r") _ r^j2_l— ^^^ ^^ 1 -r r— 1 r" — r- (4.)r (5.) ji = log- y 4- log. ^-- log- « log. r And, if the logarithm of - = N, that of = M, and tbtofrsR: then Also, if when r is a fraction, n is infinite, then is r" = 0, and the ^ipKttion for • becomes (7.) 9 ss , which expression is often of use in the summa- 1 — r tion of infinite series. h The least of ten terms in geometrical progression is 1, the '^io 2. Required the greatest term, and the sum. 88 FRACTIONAL AND NEGATIVE EXPONENTS. [PART 1. Here z = ai^~^ = 1 x ii'* = 512, the greatest term ; rz —a 2 X 512 — 1 and « =: = = 1023, the sum. r — 1 1 2. Find the sum of 12 terms of the progression 1, ^, ^, j^*-^, &c. Hei^ , = ^_, = _^pL =^^^^, the sum. 3. Find the sum of the series 1, |, ^, |, &c., carried to infinity. Here by Theor. (7.) % = , becomes * = j = ^> ^^*® sum required. 4. Find the vulgar fraction equivalent to the circulating decimal •36363636. This decimal, expressed in the form of a series, is, -^^^^ -f- T;^Jrt,) + TOoVoort + &c., where a = -^^^ , and r = -^J^ . Consequently, a = = -f^^ -r tVo = i^' '^*® fraction sought. 5. Find the sum of the descending infinite series 1 — x -r jr - J7* -f X\ &C. Here a = 1, r = — ;u, and s = = ; , the sum req. And, by way of proof, it will be found that if 1 be divided by I + ^, the quotient will be the above scries. 6. Of four numbers in geometrical progression tbc product of the two least is 8, and of the two greatest 128. What are the numbers? AnB. 2, 4, 8« and 16. Sect. XII. Fractional and Negative Exponents. In the preceding sections on powers and surds, we have only spoken of such powers as have positive and integral quantities for their exponents, and which are termed direct powers. And we ex- plained that the exponent of the quantity, expressed the number of times that that quantity was multiplied by itself; so that to denote the division of any power by its root, we have only to subtract a" 1 from tbe index or exponent of that power ; thus a" * = — ; and a" by continuing to divide by a, we have a'~'"* = a**"* = — ^, and a* a* a""^"' = a""' = — ; or generally, a"~* =— ^; that is, to express the division of one power of a qiiantitt/ by any other potter of the same quantity ^ we subtract the exponent of the divisor from that of the dividend. I eUiP. II.] FRACTIONAL AND N£OATIVK BXP0NBNT8. 89 If ft b greater than m, then will the quotient -^ = a'' have for i K^ exponent {p) a positive numher, and is then called a direct power ^-.*ta'y bat if m is greater than n, p will he negaUive^ and a"' is then a* ^r^led an inverse power of ay thus --j = a*""* = «*, is a direct a* ^>«wer, but — j- = a'~* = a~^ is an inverse power. Now, to arrive at a just idea of the value of a quantity with a x^^gadye exponent or of an inverse power, let us successively subtract mmwityfrom tlie index of any direct power; or, in other words, divide t.liftt power successively by its root ; thus, let a" be the direct nS^ pcwerofo, then «— ' = a' a' a»-' = T-- a»-' = ^ = . = . «'-' = °'='='. a a 1 1 1 1 a <r "* a? • "* a* "^ ' ^, generally, tf~* = — . TheU is^ the value of an inverse potg>er Septal to unitff divided by the same direct power. As the properties of inverse powers are the same as those of direct Jowen, all that has been stated in Section V. regarding the latter, ^oally applies to the former, and it is therefore unnecessary to recapitulate it here. We have, however, yet to mention another kind of exponent, viz., the fractional, used to express the roots of quantiUes, in a similar manner to that employed for expressing powers ; and which extends to them also, all the properties already stated as belonging to inverse tnd direct powers. This method of expressing any root of a quantity, consists in ittaching to that quantity as an exponent, a fraction having unity for its numerator, and the index of the root for its denominator. i 90 LOGARITHMS. [pABT I. Thus \/a is written or 1 i_ And the roots thus expressed may be employed in algebraical cal- culations in the same manner as powers with integral exponents. Thus, if we wish to express the square of the cube of jr, we do so by x'^ ^ - = a/* ; and in like manner, to express the square of the cube root of ;r, we should have j?l><* =: x'; or, the square of the square root of X =. A^^ z:^ xi =^ X itself, which is obvious. And in It general, x" may be employed to express the n^ power of the ni^^ root of J*, or what is the same thing, the m^^ root of the n*^ power of X. If the fractional exponent has a negative sign, its value will be found by the rule already given for integral inverse powers; thus, -" 1 1 «'" = — = ;;;— r^ = Unity dividcd bv the m}^ root of the w'** power of a. It only remains to state, that in using fractional exponents, we may substitute any equivalent fraction, or a decimal, in place of the original exponent; thus a-, a*, a**, a*^, are only so many different ways of expressing the square root of a; advantage may frequently be taken of this circumstance, to facilitate the working with surds. By way of recapitulation, then, there are four different kinds of exponents employed in analytical operations, viz. : — The posittce iyitegral exponent, as .r", which denotes the direct nth power of a?, and is equal to x multiplied w times by itself. The ne(fative integral exponent, as a; " ", which denotes the inverse n^^ power of ar, and is equal to unity divided n times by x. The positive fractional exponent, as «", which denotes the direct n*** root of ;r, and is equal to a quantity, which being multiplied n times into itself will equal x. The negative fractional exponent, as x~ » , which denotes the in- verse n^^ root of a;, and is equal to unity divided by the direct n^ root of X, Sect. XIII. Logaritfnns. LooARiTHiis are a series of numbers in arithmetical progression, answering to another series in geometrical progression ; so ti^en that in the former corresponds with 1 in the latter. CHAP. II.] LOGARITHMS. 91 Thus, 0, 1, 2, 3, 4, 5, are the Logs, or aritkmetieai series ; and 1, 2, 4, 8, 16, 32, | ^'^ ^^^ ^^f ' ^' 9e(nnetricd series, an- ' ' ' ' ' 'J swenng thereto. Or, 0, I, 2, 3, 4, 5, the logarithms ; and J, 5, 25, 125, 025, 3125, the corresponding numbers. Or, 0, 1, 2, 3, 4, 5, the logarithms ; and 1, 10, 100, 1000, 10000, 100000, the corresponding numbers. In which it will be seen, that by altering the common ratio of the geometrical series, the same arithmetical series may be made to serve as the logarithms of any series of numbers. As above, where the common ratios of the several geometrical series are 2, 5, and 10, re- spectively. Or, the logarithms of a number may be considered as the indices or exponents of the powers of some root or radix, which when in- volved to the power indicated by the logarithm, is then equal to the number. Thus, taking the foregoing series as an illustration, in which the roots or radices are 2, 5, and 10, respectively, we have Nos. Logs, Nos. Logs. Nos. JLog». 1 = 2 « i = 5 " 1 = 10 " 2 = 2' 5 = 6* 10 = 10 * 4 = 2 =* 25 = 5 * 100 = 10 « 8 = 2'* 126 = 5 •'* 1000 = 10 » 16 = 2 ^ 625 = 5 ^ 10000 = 10 * 32 = 2 * 3126 = 5 * 1(M)000 = 10 * in which the exponents in the columns headed Xo^«., are the logar- rithms of the numbers in the first column headed Nos. The last of these series, or that in which the common ratio of the geometrical series, or (what is the same thing) the radix of the system of logarithms is 1 0, is that usually employed for the purpose of facilitating calculations ; and is the same as the series of logarithms contained in the Logarithmic Tables in the Appendix to this work. As in this system of logarithms the numbers whose logarithms are integral increase as the powers of 10, and it is requisite in calcula- tions to use the logarithms of the intermediate numbers, it becomes necessary to interpolate with fractional logarithms ; thus the series then becomes Nos. Logi. Nos. Loot. 1 = 10 •• 8 = 10 •*>** 2 = 10 •«"« 9 = 10 •«*«» 3 = 10 *"^'"" 10 = 10 »• 4 = 10 •«»• 11 = 10 «•«««« 6 = 10 «■«»»' 12 = 10 »•»»•« 6 = 10 *"»»^» IS = 10 ••'«»" 7 = 10 "^^^ &c. &c. in which the exponents of the powers of 10, answering to any given uamber, will be found to correspond with the logarithm of that num- ber as given in Table H. in the Appendix. 92 LOGARITHMS. [PABT I. In order to explain the properties oF logarithms, and the reason of the rules given in the section on Logarithmic Arithmetic, let as as- sume any series of numhers, N, N^, Ng, N^, &c., to which corre- spond the logarithms, /, l^y Ly l^y &c., to the same root or radix (r). 1. Then we have, N = r*, Nj = r'*, N^ = r\ &c. ; and by the rule for multiplpng powers, (page 66,) r'xr'» = r'^'>=Nx N,; and N.Nj .Ng =r'+'» + '«. Or the logarithm of the product oj two or more numbers^ is equal to the sum of the logarithms of those numbers; and conversefyy the logarithm of the quotient of two num^ berSj is equal to the difference of the loaarithms of the dividend and divisor ; which correspond with the rules given at page 45, for the multiplication and division of numbers by logarithms. 2. IfN = Nj =No = N3; then we have, by the foregoins, log (N.N.N .N)= log (N') = r'+ '+' + ' = r*'; and in genenU, log (N") = r"'. Or the logarithm of the n** power of any quaniitj/ is equal to n times the logarWim of thai quantity^ whether n is int^al, fractional^ negative^ or positive. For let A be the quantity, and let n be negative ; then A"" = — ;, and log ( — ) = ^ogof I ^ n (log A) ; and as log of 1 =: 0, log 1 — » log A = — » (log A) = log ( A-"). P - If n = — ; then, let A ? = K ; raising both sides to the power ^, we have A'' = K^, and their logarithms, p (log A) = ^ (log K) ; then P - dividing both sides by 7, we obtain - (log A) = log K = log (Al). p -t I L Lastly, if » = — -, then will A « = — ; and since log (A •) =a ^ a5 |(log A) .-. log/i \= logl -^ (logA)= - ^ (logA) = log(A--0- 8. In Section XI., page 41, it was stated that the decimal portion of a loearithm was always the same for the same digits, and that ii was only the characteristic which was altered, by the multiplication or division of the quantitv by 10, or any power of 10 ; the reason of this is obvious, since all the powers of 10 have integers only foi their logarithms. 4. It was also stated, (page 41,) that it was only the characteristic of the logarithm of a firaction, which was negative, the decimal portion being always positive; to illustrate this, let us take the fraction -05 = -^ = — -j— = lO"* x lO**"*^** = 10 - 2+ -cwwo . and therefore 2-698970 is the log. of -05. 5. Suppose there be two systems of logarithms whose roots or CHIP. II.] LOGARITHMS. 93 bises are r and «. Let any number N have p for its loearithm in the first system, and q for its logarithm in the second : we shidl have N = r^ and N = I* ; which gives »* = #», and * = r». Therefore, taking the logarithms for the system r, we shall have log 9 =^(logr); or, if in the system r we have log r = 1, then log » = -, or y = pi l£- =s p X . . Thus, knowing the logarithm p of any number N, for the system whose base is r, we may obtain the logarithm q of tlie same number for the system «, by multiplying je? by a fraction vlioee numerator is unity and denominator the logarithm of 8 taken Id the sjstem r. 6 In the system of logarithms first constructed by Baron Napier, the great inventor, 9 = 2*718281828459, &c., and the exponents ire usually denominated Napierian^ or Hyperbolic logarithms; the latter name beins given because of the relation between these log- triihms and the lines and asymptotic spaces in the equilateral hyper- bola: 80 that in this system n is always the hyperbolic logarithm of (2-718*28, &c.)*. But in the system constructed by Mr. Briggs (correspondm^ with the spaces in a hyperbola whose asymptotes make an angle of 25° 44' 25" 28'"), called common or Briggean logarithms, r = 10 ; so that the common logarithm of any number is, as already stated, the index of that power of 10 which is equal to the said number. 7. Although the Briggean logarithms are those usually employed for the purpose of computation, the hyperbolic are always used in the differentia] calculus, and the higher branches of analysis ; it there- fore becomes somedmes requisite to find the hyperbolic logarithm of a quantity, which may be obtained from the common logarithm as follows : from § 5, above, we have q =: p -. ; then, * putting ^' log for hyperbolic logarithm, log for common logarithm, and sub- «itating for * its proper value, we have h. log n = log n -. ^717757^^7 log 2'71o281 = log n Q.^3^^g^^Q = 2-3025851 (log «) ; or io obtain the hyper- Ik^Hc logarithm of any number^ multiply its conmion logarithm by 2-S025851. In practice the following method will be found more convenient tlam multiplying by 2-3026851. To convert common logs, into hyperbolic. Write the common log, as shown in the following Examples, and then ^e from Table I. the equivalent value of each figure 10 hyperbolic 1^ taking care that the latter are each moved as many places to the 'ight as the corresponding numbers in the common logarithm. The 8nm of the whole will then be the hyperbolic logarithm required. 1)4 LOGARITHMS. [part I. To convert hypei-hdic logs, into common. Proceed in the same manner, only using Table II. instead of Table I. TABLE I. TABLE XL S: «yp-^^»- Hyp. Logs. Com. Logs. 1- 2-3025851 2- 4-6051702 3- , 6-9077553 4- 9 2103404 5- 1 11-5129255 6- ' 13-8155106 7- 161180957 8- , 18-4206807 9- 1 20-7232658 1 2- 3- 4- 5- 6- 7- 8- 9^ •4342945 •8685890 1-30-28834 1-7371779 21714724 2-6057669 3-0400614 3-4743559 3-9086503 Examples. I. What is the hyp. log. of 1662 ? By reference to Table III. in the Appendix, we find that 3-220631 is the common log. of 1662 ; then 7-415778? Cora. Log. Hyp. Log. 3-000000 = 6-907755 3 •200000 = -460517 •020000 = •046051 7 -000600 = •001381 5 -000030 = -000069 •000001 log. ( •000002 7-415778 3 3-220631 i common )f the numb er Hyp. Log. Com. Log , 7-000000 ^ 3-040061 4 •400000 = -173717 7 •010000 = •004342 9 -005000 ^ •002171 4 •000700 ^ -000304 -000070 = •000030 4 -000008 = •000003 4 7-416778 3220631 CHAP. II.] COMPUTATION OP PORMULiG. 95 Sect. XIV. ComptUcUion of FormulcB. Since the comprehension, and the numerical computation of for- iiu]« expressed aJgebraically, are of the utmost consequence to prac- ical men, enabling them to avail themselves advantageously of the Lbcorelical results of men of science, as well as to express in scien- tific language the results of their own experimental or other re- semrthes; it has appeared expedient to present brief treatises of Aiitbinetic and Algebra. The thorough understanding of these two initiatory departments of science will serve essentially in the applica- QOD of all that follows in the present volume ; and that application mar probably be fieusilitated by a few examples, as helow : — £i. I. Let 6 = 5, <r = 12, d = 13, and s = ; then what Kthe numerical value of the expression \/« («— ^) • (* — c) . (« — </) from page 1 27, which denotes the area of the triangle whose sides are a, 12, and 18? „ 6-hc + J 5 + 12 + 13 ,^,, . Here i = — ^^-^^ — = — ■ — ^ = 15 ; then « — ^ = 15 -5 = 5; <~c= 15 — 12 = 3; and« — </= 15 — 13 = 2. CoDseqaently, by substituting the numerical values of the several qoMtities between the parentheses for them, we shall have v/(l5 X 10 X 3 X 2) = x/900 = 30, the value required. The same values being given to 6, c, and c?, we may, as a verifica- tiOQ of the above, compute the area of the same triangle from the eqaiTalent expression, /\J ^ - ( ^^ "^ 2/ ' 2* Herer = 12- = 144 ; 6- = 5- = 25 ; 2rf = 26; and ^ = 6|. Sabstituung these, the expression becomes the same result as before. ^J?. 2. Suppose ^ = 32^, < = 6 : required the value of J g f, •D expression denoting the space in feet which a heavy body would Wl TerticaUy from quiescence in six seconds, in the latitude of I<ondon. Here |^^ = 16-^^ x 6* = 96i x 6 = 579 feet. ^3. Given D = 6, rf = 4, A = 12, a = 3*141593 ; required the »iloeof y^ «^A(D' + D</ + d% a theorem for the solid content of * conic fhwtam whose diameters of the two ends are D, </, and lieigbtA. 9fi COMPUTATION OF FORMULA. [PART I. Here D- = 36, D r/ = 6 x 4 = 24, d' =16,^ = -2618 nearly. Hence -jV v A (D* + D<f + d') = -2618 (36 + 24 -f 16) 12 = •2618 X 76 X 12 = 3141593 x 76 = 238-761068. Ex, 4. Let a = I, A = 25, ^ = 193 inches : what is the value of 2 a ^g h ? This heing the expression for the cuhic inches of water discharged in a second, from an orifice whose area is a, and depth below the upper surface of water in the vessel, or reservoir, A, both in inches. Here2av^^A=:2 v^(25 x 193) = 10 >/193 = 10 x 13-89244 = 138*9244 cubic inches. Ex, 5. Suppose the velocity of the wind to be known in miles per hour ; required short approximative expressions for the yards per minute, and for the feet per second. First 1760 ^ 60 = *y^ = 29 J = 30 nearly. Also 5280 -h (60 x 60) = f^jo = gg = 4^ = 1 J nearly. If, therefore, n denote the number of miles per hour : 30 n will express the yards per minute; and 1|«, the feet per second. These are approximative results : to render them correct, where complete accuracy is required, subtract from each result its 45th part, or the Ji/lh part of its ninth part. Thus, suppose the wind blows at the rate of 20 miles per hour : Tlien 30 n = 30 X 20 = 600 yards per minute, or more cor- rectly 600 — Y:^ = 600 — 13i = 6865 yards. Also 1 J w = 30 feet per second ; or, correctly 30 — ^ J = 30 — ^ = 29^ feet Conversely, f of the feet per second will indicate the miles per hour, correct within the 45th part, which is to be added to obtain the true result. Ex. 6. To find a theorem by means of which it may be ascertained when a general law exists, and what that law is. Suppose, for example, it were required to determine the law which prevailed between the resistances of bodies moving in the air and other resisting media, and the velocities with which they move. Let V,, V.^, denote any two velocities, and R,, R.,, the corresponding resistances experienced by a body moving with tliose velocities : we wish to ascertain what power of V , it is to which R j is propor- tional. Let X denote the index or exponent of the power : then will V* : Vlj : : Rj : R^, if a law subsist. CHAP. II.] COMPUTATION OF FORMULiE. 97 (V \' R — I : : 1 : — ^ . (Vo\' R — ^ I = — ^. This, expressed logarithmically, gives * X log^=log^; ^^^^logR^_-_logR, logV, - logV/ Hence the quotient of the difTerences of the logs, of the resistances, divided hy the difference of the corresponding velocities, virill express the exponent z required. This theorem is of very frequent application in reference to the motion of cannon balls, of barges on canals, of carriages on rail- roads, &c., and may indeed be applied to the planetary motions. When two or more values of any quantities occur in a formula, it is usual to denote both of them by the same letter, distinguishing between them by the attachment of a small number below it, as in the last example above, where the two velocities are both denoted by the letter V, but distinguished by the numerals, as Vj and V^, and the two resistances by R^ and R^. 08 OKOMETRICAL DBF1NITI0N8. [PART I. CHAP. III. PLANE AND SOLID GEOMETRY. Sect. I. Defi?iitions. I. Geometry is that departmcDt of science, by means of which we (Icnionstrate the properties, affections, and measures of all sorts of ma^7iitude, ii. Magnitude is a term used to denote the extension of any thing, and is of three kinds ; as the magnitude of a /twe, which is only in one direction, viz. length ; the magnitude of a surface^ which is in tvo directions, viz. length and breadth; and the magnitude of a solid^ which is in three directions, viz. lengthy breadth and depth, 3. A ]X)int has no parts or magnitude ; neither length, breadth, nor thickness, and serves only to assign position, 4. A line has length without breadtli or thickness, and indicates direction and distance. Cor, The extremities of a line are, therefore, points. 5. A right line is that which lies evenly, or in the same direction, between two points. A curve liyie continually changes its direction. Cor, Hence there can only be one species of right lines, but there is an infinite variety in the species of curves. 6. An angle is the inclination of two lines to one another, meeting in a point, called the angular point When it is formed by two rig^ lines^ it is a plane angle, as A ; if by curve lines, it is a curvUineal angle, as B. (Fig. 1.) 7. A right angle is that which is made by one right line A B fall- ing upon another C D, and making the angles on each side equal, that is, A B C = A B D ; so that the line A B does not incline more to one side than another : it is then said to be perpendicular to the line C D. All other angles are called oblique angles. (Fig. ^.) 8. An obtuse angk \s greater than a right angle, as R. (Fig. 3.) 0. An acute angle is less than a right angle, as S. (Fig. 3.) 10. Contiguous or adjacent angles^ are the two angles formed by one line falling upon another, as R, S. (Fig. 3.) I I . Vertical or opposite angles, are those made on contrary sides of two lines intersecting one another, as A £ C, D £ B. (Fig. 4.) 12. A surface has length and breadth, but no thickness, and shows extension. The area of a figure is the quantity of space which its surface occupies. CHAP. III.] ANGLES, AND BIGHT LINES. 99 Cor, The boundaries or limits of a surface are lines. 13. A plane is that surface which lies perfectly even between its extremes; or which, being cut by another plane in any direction, its section would be a straight line. 14. A idid is a magnitude extended every way, or which has length, breadth, and depth. Cor, The boundaries or extremes of a solid are surfaces. 15. The square of a right line is the space included by four right lines equal to it, set perpendicular to one another. 16. The rectangle of two lines is the space included by four lines equal to them, set perpendicular to one another, the opposite ones being equal. 17. One right line is said to he parallel io another, when both lines, being extended to any length, will never meet, but always preserve the same distance between them. Sect. II. Of Angles^ and Right Lines^ and their Rectangles, Prop. I. If to any point C in a right line A B, several other right lines D C, £ C are drawn on the same side ; all the angles formed at the point C, taken together, are equal to two right angles ; thus, ACD + DCE + ECB = the two right angles A C P + P C B. (F%. 5.) rEuclid, Book I. Prop. 13.] Cor, 1. AH the angles made about one point in a plane, being taken together, are equal to four right angles. Cor. 2. If all the angles at C, on one side of the line A B, are found to be equal to two right ancles ; then A C B is a straight line. (Fig. 5.) [Euc. B. I. Prop. 14.J II. If two right lines, A B, CD, cut one another, the opposite angles C £ B and A £ D will be e^ual. (Fig. 4.) [Euc. B. I. Prop. 15.] III. A right line, B I, which is perpendicular to one of two pa- rallels, is perpendicular to the other. (Fig. 6.) IV. If a right line C G, intersects two parallels AD, F H ; the alternate angles, ABE, and B £ H, will be equal. (Fig. 6.) QEuc. B. I. Prop. 29.] Cor. 1. The two internal angles D B £ and B E H on the same side are equal to two right angles. (Fig. 6.) Cot. d. The external angle C B D, is equal to the internal angle on the same side B £ H. ^Fig. 6.) [Euc. B. I. Prop. 28.] V. Right lines, parallel to the same right line, are parallel to one another. [Euc. B. I. Prop. 80.] VI. If a right line A G be divided into two parts A B, B C ; the square of the whole line is equal to the squares of both the parts, and twice the rectangle of the parts ; or A C* = A B* + B C'^ + 2 A B x BC. (Pig. 7.) [Euc. B. II. Prop. 4.] VII. The square of the difference of two lines A C, B C, is equal to the sum of their squares, wanting twice their rectangle ; or A B^ ^ AC" + BC»— 2AC X BC. OrAB* -f 2AC X BC = AC* -f B C*. (Fig. 7.) [Euc. B. II. Prop. 7.] H 2 100 TRIANGLES. [PABT I. VIII. The square of the sum of two lines is equal to the sum of their squares, together with their rectangle. IX. The rectangle of the sum and difference of two Hoes is equal to the difference of their squares. X. The square of the sura, together with the square of the differ- ence of two lines, is equal to twice the sum of their squares. Sect. III. Of Triangles. DEFINITIONS. 1 . A triangle is a plane figure bounded by three right lines^ called the sides of the triangle. 2. An equilateral triangle is one in which all the three sides are equal. 3. An isosceles triangle has only two sides equal. 4. A scalene triangle has all its three sides unequal. 5. An equiangular triangle is one which has three equal angles; and two triangles are said to be equiangular, when the angles in the one are respectively equal to those in the other. 6. A right-anghd triangle is that which has one of its angles a right angle. The side opposite to the right angle is called the hypotenuse^ and the other two sides the legs. 7. An oUique angled triangle has all its angles oblique. 8. An obtuse angled triangle has one of its angles obtuse. 9. An acute angled triangle has all its angles acute. 10. In a triangle, the lowest side, as A B, is called the hase^ and the opposite angle C the vertex : the altitude of a triangle is the per- pendicular height (CD) from the base, or its extension to the yertex. (Fig. 8.) 1 1. Similar triangles are those whose angles or sides are Tesped^ ively equal, each to each. And homologous sides are those lying between equal angles. PROPOSITIONS. Prop. I. In any triangle ABC, if one side B C be produced or drawn out ; the external angle A C D will be equal to the two inter- nal opposite angles ABC and BA C. (Fig. 9.) [Euc. B.I. Prop. 32.1 II. In any triangle, the sum of the three internal angles is equal to two right angles. [Euc. B. I. Prop. 32.] Cor, 1. If two angles in one triangle be equal to two angles in another : the third will also be equal to the third. Cor, 2. If one angle of a triangle be a right angle, the sum of the other two will be equal to a right angle. III. The angles at the base of an isosceles triangle, are equal. [Eiic. B. I. Prop. 5.] Cor, 1 . An equilateral triangle is also equiangular ; and the con- trary. Cor. 2. The line which is perpendicular to the base of an iaosceles triangle, bisects both it and the yertical angle. CHIP, in.] TRIANGLES. 101 IV. In any triaogle, the greatest side is opposite to the greatest ugle, and the least to the least. [Euc. B. I. Prop. 18.] V. In any triangle ABC, the sura of any two sides B A, A C, is greater than the third B C, and their difTerence is less than the third ade. (Fig. 10.) [Euc. B. I. Prop. 20.] VI. If two triangles ABC, a 6 c, have two sides, and the included togle equal in each ; these triangles, and their correspondent parts, M be equal. (Fig. 11.) [Euc. B. I. Prop. 4.] VII. If two triangles ABC and abc^ have two angles and an in- cloded side equal, each to each ; the remaining parts shall he equal, •ndthe whole triangles equal. (Fig. 11.) [Euc. B. I. Prop. 26.] VIII. Triangles of equal bases and heights are equal. [Euc. B. I. Prop. 37.] IX. Triangles of the same height, are in proportion to one another as their bases. [Euc. B. VI. Prop. 1.] X. If a line D E be drawn parallel to one side B C, of a triangle ; the segments of the other sides will be proportional ; that is, AD:DB:: AE : EC. (Fig. 12.) [Euc. B. VI. Prop. 2.] Cor, 1. If the segments be proportional, A D : D B : : A E : EC ; then the line D £ is parallel to the side B C. (Fig. 12.) C<fr. 2. If several lines be drawn parallel to one side of a triangle, >I] the segments will be proportional. Cor. 3. A line drawn parallel to any side of a triangle, cuts off a triiDgle similar to the whole. XI. Id similar triangles, the homologous sides are proportional ; thatis,AB : AC :: DE : DF. (Fig. 13.) [Euc. B. VI. Prop. 4.] XII. Like triangles are in the duplicate ratio, or as the squares of, their homologous sides. [Euc. B. VI. Prop. 19 ] XIII. In any triangle, the difference of the squares of the two legs A Band A C, is equal to twice the rectangle contained by the base Be, and the distance DO, of its middle point from the perpendicu- lar DA. (Fig. 14.) XIV. In a right-angled triangle B A C, if a perpendicular be let fill from the right angle upon the hypothenuse, it will divide it into two triangles, similar to one another and to the whole, A B D, A D C. (rig.l4.) [Euc. B. VI. Prop. 8.] Cor, 1. The rectangle of the hypothenuse and either segment is eqoal to the square of the adjoining side. XV. The distance A O of the right angle, from the middle of the hjpotbenuse is equal to half the hypothenuse. (Fig. 14.) XVI. In a right-angled triangle, the square of the hypothenuse is eqwd to the sum of the squares of the two sides. [Euc. B. I. Prop. 47.] XVII. If the square of one side of a triangle be equal to the •om of the squares of the other two sides ; then the angle compre- ^ed by them is a right angle. [Euc. B. I. Prop. 48.] XVIII. I fan angle A, of a triangle B A C be bisected by a right be A D, which cuts the base ; the segments of the base will be Proportional to the adjoining sides of the triangle; that is, BD: DC :: AB : AC. (Fig. 10.) [Euc. B. VI. Prop. 3.] 102 QUADRILATERALS AND POLYGONS. [PABT I. XIX. If the sides be as the segments of the base, the line A D bisects the angle A. (Fig. 10.) [Euc. B. VI. Prop. A.l XX. Three lines drauTi from the three angles of a triangle to the middle of the opposite sides, all meet in one point. XXI. Three perpendicular lines erected on the middle of the three sides of any triangle, all meet in one point. XXII. The point of intersection of the three perpendiculars, will be equally distant from the three angles ; or, it will be the centre of the circumscribing circle. XXIII. Three perpendiculars drawn from the three angles of a triangle, upon the opposite sides, all meet in one point. XXIV. Three lines bisecting the tliree angles of a triangle, all meet in one point. XXV. If D be any point in the base of a scalene triangle, ABC: then is A B- X D C -f A C- x B D = AD^ xBC-fBCxBD X DC. (Fig. 10.) Sect. IV. Of QuadrUatercdB and Pdi/gons, DEFINITIONS. 1. A quadrangle or quadrilaieral^ is a plane figure bounded by four right lines. 2. A paraUelogram is a quadrangle whose opposite sides are pa- rallel, as AGBH. The line A B drawn to the opposite corners is called the diameter or diagonal. And if two lines be drawn parallel to the two sides, through any point of the diagonal, they divide it into several others, and then C and D are called parallelograms about the diameter; and E and F the complements; and the figure £ DF a gnomon, (Fig. 15.) 3. A rectangle is a parallelogram whose sides are perpendicular to one another. 4. A square is a rectangle of four equal sides and four equal angles. 5. A rhombus is a parallelogram, whose sides are equal, and all its angles oblique. 6. A rhomboid is a parallelogram in which only the opposite sides arc equal, and all its angles oblique. 7. A trapezoid is a quadrangle, having only two sides parallel, as Fig. IG. 8. A trapeziuyn is a quadrangle that has no two sides parallel, as Fig. 17. 9. A pdygon is a plane figure enclosed by many right lines. If all the sides and angles are equal, it is called a regular polygon, and denominated according to the number of sides or angles, as a pen- tagon having five sides, a hexagon^ having six sides, a heptagon^ having seven sides, &c. 10. The diagonal of a quadrangle or polygon is a line drawn be- tween any two opposite comers of the figure, as A B. (Fig. 15.) diP.ni.] QUADBI LATERALS AND POLYGONS. 103 11. The height of a figure is a line drawn from its vertex^ perpen- ^licular to the base* or opposite side on which it stands. 12. Like or nmiiar figures, are those whose several angles are •equal to one another, and the sides about the equal angles proper- lioQiL 13. Homelogmte sides of two like figures are those between two angles, respectively equal. 14. The perimeter or circumference of a figure, is the compass of it, or sum of all the lines that enclose it. 15. The internal angles of a figure are those on the inside, made l»y the lines that bound the figure, as A B C, D C B, &c. (Fig. 18.) 16. The external angles of a figure are the angles made by each lide of a figure, and the adjoining side drawn out, as B A F, A £ G. (Fig. 18.) PROPOSITIONS. Pbop. I. In any parallelogram the opposite sides and angles are equal ; and the diagonal divides it into two equal triangles. [Euc. B. I. Prop. 34.] II. The diagonals of a parallelogram intersect each other in the middle point of both. III. Any line B C passing through the middle of the diagonal of » parallelogram P, divides the area into two equal parts. (Fig. 19.) IV. Any right line B C drawn through the middle point P of the diigooal of a parallelogram, is bisected in that point ; or B P = P C. V. In any parallelogram A G H B, the complements £ and F are equal. (Fig. 15,) [Euc. B. I. Prop. 43. J VI. Parallelograms of equal bases and heights are equal. [Euc. B. I. Prop. 36.] VII. The area of a parallelogram is double the area of a triangle binngthe same base and height. [Euc. B. I. Prop. 41.] VIII. Parallelograms of the same height are to one another as tbeir bases. [Euc. B. VI. Prop. 1.] IX. Parallelograms of equal bases are as their heights. X. Parallelograms are to one another, as their bases and heights. XI. In any parallelogram the sum of the squares of the diagonals ii eqoal to the sum of the squares of all the four sides. XII. The sum of the four internal angles of any quadrilateral %ore, is equal to four right angles. XIII. If two angles of a quadrangle be right angles, the sum of tlie other two amounts to two right angles. XIV. The sum of all the internal angles of a polygon is equal to tvice as many risht angles, abating four, as the polygon has sides. Cor. Hence all right-lined figures of the same number of sides, hate the sum of all the internal angles equal. Xy. The sum of the external angles of any polygon is equal to four right angles. Cor, All right-lined figures have the sum of their external angles eqoal. 104 CIRCLBP, ETC. [part I. XVI. In two similar figures AC, PR; if two lines BE, Q T, be drawn after a like manner, as suppose, to make the angle C B E = R Q T ; then these lines liave the same proportion as any two homologous sides of the figure ; viz., BE:QT::BC:QR::AB:PQ::AD:PS. (Fig. 20.) XVII. All similar figures are to one another as the squares of their homologous sides. XVIII. Any figure described on the hypothemise of a right- angled triangle, is equal to two similar figures described the same way upon the two legs; that is, BFC=ALC -fAGB. (Fig. 21.) [Euc. B. VI. Prop. 31.] XIX. Any regular figure ABCDE, is equal to a triangle whose base is the perimeter A B C D E A ; and height the line O P, drawn from the centre, perpendicular to one side. (Fi<r. 22.) XX. Only three sorts of regular figures can fill up a plane surface, that is, the whole space round an assumed point, and these are six triangles, four squares, or three hexagons. Sect. V. Of the Circle^ and Inscribed and Circumscribed Figures, DEFINITIONS. 1. A circle is a plane figure described by a right line moving about a fixed ]>oint, as A C about C : or it is a figure bounded by a curved line, every part of which is equidistant from a fixed point. (Fig. 23.) 2. The centre of a circle is the fixed point about which the line moves, as C. (Fig. 23.) 3. The radius is the line that describes the circle, as C A. (Fig. 23.) Cor, All the radii of a circle are equal. 4. The circumference is the line described by the extreme end of the moving line, as A B D E A. (Fig. 23.) 5. The diameter is a line drawn through the centre, from one side to the other, as A D. (Fig. 23.) 6. A semicircle is half the circle, cut off by the diameter, as ABD. 7. A quadrant,, or quarter of a circle, is the part between two radii perpendicular to one another, as C DE. (Fig. 23.) 8. An arc is any part of the circumference, as A B. (Fig. 24.) .9. A sector is a part bounded by two radii, and the arc between them, as A C B. (Fig. 24.) 10. A chord is a right line drawn through the circle, as D F. 1 1. A segment is a part cut off by a right line, or chord, as D £ F, or DABfI (Fig. 24.) 12. Angle at the centre is that whose angular point is at the centre A C B. (Fig. 24.) 13. Angle at the circumference is when the angular point is in the circumference, as BAD, or BCD. (Fig. 25.) 14. Angle in a segment^ is the angle made by two lines drawn CHAP. III.] CIRCLES, ETC. 105 irom some point of tbe arc of that segment to the ends of the base ; IS BCD is an angle in the segment BCD. (Fig. 25.) 15. Angle upon a segment is the angle made in the opposite Kgment, whose sides stand upon the base of the first ; as BAD, which stands upon tlie segment BCD. (Fig. 25.) 16. A tangent is a line touching a circle, which, produced, does sot cat it, as 6 A F. (Fig. 23.) IT. Circles are said to touch one another, which meet, but do not eat one another. 18. Similar arcs, or similar sectors^ are those bounded by radii that make the same angle. 19. Similar segments are those which contain similar triangles, ilike placed. 20. A figure is said to be inscribed in a circle, or a circle circum- mhed about a figure^ when all the angular points of the figure are in tbe circumference of the circle. (Fig. 26.) 21. A circle is said to be inscribed in a figure, or a figure circum^ Knhed about a circle, when the circle touches all the sides of the figure. (Fig. 27.) 22. One figure is inscribed in awo/Aer, nvhen all the angles of the mscribed figure are in the sides of the other. (Fig. 28.) PROPOSITIONS. Phop. I. The radius C R, bisects any chord at right angles, which does not pass through the centre, as A B. (Fig. 29.) [Euc. B. III. Prop. 3.] Cor, 1. If a line bisects a chord at right angles, it passes through tie ccDtre of the circle. Cor. 2. The radius that bisects the chord also bisects the arc. II. In a circle equal chords are equally distant from the centre. [Edc. B. III. Prop. 14.] III. If 8e?eral lines be drawn through a circle, the greatest is the diaineter, and those that are nearest the centre are greater than those tbtt are farther off. fEuc. B. III. Prop. 15.] IV. If from any point three equal right lines can be drawn to the ojcamference, that point is the centre. [Euc. B. III. Prop. 9.] V. No circle can cut another in more than two points. [Euc. B. III. Prop. 10.] yi. There can be only two equal lines drawn from any exterior point, to the circumference of a circle. VIL In any circle, if several radii be drawn making equal angles, tie arcs and sectors comprehended thereby will be equal ; that is, if tbe angle A C B = B C D, then, the arc A B = B D, and the sector ACB = BCD. (Fig. 30.) [Euc. B. III. Prop. 26.] VIII. In the same or equal circles, the arcs, and also the sectors, ve proportional to the angles intercepted by the radii. IX. The circumferences of circles are to one another as their diuneters. X. A right line, perpendicular to the diameter of a circle, at the 106 CIBCLBS, ETC. [PART I. extreme point, toncbes die circle in that point, and lies wholly withoot the circle ; or is a tangent to the circle. XI. If two circles- touch one another, either inwardly or outwardly, the line passing throogh their centres shall also pass through the point of contact. [Euc. B. III. Prop 11 and 12.] XII. In a circle the angle at the centre is double the angle at the circumference, standing upon the same arc; orBDC = 2BAC (Fi*r. 31.) Euc. B. III. Prop. 20.] XIII. All angles in the same segment of a circle are equal, as DAC = DBC, and DOC=DHC. (Fig 32.) [Eac. B. III. Prop. 21.; XIV. if two right lines DC, A B, be drawn from the extremities of two equal arcs D A, B C, they will be parallel. (Fig. 32.) XV. The angle A B C in a semicircle is a right angle. (Fig. 33.) [Euc. B. III. Prop. 31.; XVI. The angle FBO, in a greater segment FABCO, is less than a right angle ; and the angle D B E, in a less segment D B E, is greater than a right angle. (Fig. 33.) [Euc. B. III. Prop. 31.] X VII. If two lines cutting a circle, intersect one another in A ; and there be made at the centre, Z.ECF = Z.BAD; then the arc B D 4- G H = 2 E F, if A is within the circle ; or the arc B D — G H == 2 E F, if A is without. (Fig. 34.) XVIII. If from a point without, two lines, A B, AD be drawn to cut a circle ; the angle made by them is equal to the angle at the centre, standing on half the difference of the two arcs of the circum- ference G H, B D. (Fig. 31..) XIX. The angle A = Z.BHD-f-HDO, when A is witliin ; or A = B H D — H D G, when A is without the circle. (Fig. 34.) XX. In a circle, the angle made at the point of contact between the tangent and any chord, is equal to the angle in the alternate or opposite segment; ECF=EBC, and ECA=EGC. (Fig. 35.) [Euc. B. III. Prop. 32.] XXI. A tangent to the middle point of an arc, is parallel to the chord of it. XXII. If from any point B in a semicircle, a perpendicular B D be let fall upon the diameter, it will be a mean proportional between the segments of the diameter; that is, AD : DB :: DB : DC. (Fig. 30.) [Euc. B. VI. Prop. 13.] XXIII. The chord is a mean proportional between the adjoining segment and the diameter, from the similarity of the triangles : that is, A D : A B :: A B : AC ; and C D : CB :: C B : CA. (Fig. 36.) XXIV. In a circle, if the diameter A D be drawn, and from the ends of the chords A B, AC, perpendiculars be drawn upon the diameter ; the squares of the chords will be as the segments of the diameter ; that is, A E : A F :: A B' : AC". (Fig. 37.) XXV. If two circles touch one another in P, and the line PD£ be drawn through their centres ; and any line P A B is drawn through that point to cut the circles, that line 'will be divided in proportion to the diameters; that is PA : PB :: PD : PE. (Fig. 38.) CHlP.m.] CIRCLES, ETC. 107 XXVI. If through any point F in the diameter of a circle, any chord, C F D be drawn, the rectangle of the segments of the chord is eqaal to the rectangle of the segments of the diameter ; C F x F D = AF X FB = also GF x FE. (Fig. 39.) [Euc. B. III. Prop. 35.] XXVII. If through any point F out of the circle in the diameter BA produced, any line F C D be drawn through the circle : the rect- ingle of the whole line and the external part is equal to the rectangle of the whole line passing through the centre, and the external part ; DFxFC = AF X FB = also F E X F G. (Fig. 40.) XXVIIL Let H F be a tangent at H ; then the rectangle C F x FD = square of the tangent F H. (Fig. 40.) [Euc. B. III. Prop. 36.] XXIX. If from the same point F, two tangents be drawn to the circle, they will be equal ; that is, F H = F I. (Fig. 40.) XXX. If a line P F C be drawn perpendicular to the diameter AD of a circle; and any line drawn from A to cut the circle and the perpendicular ; then the rectangle of the distances of the sections from Ay will ^be equal to the rectangle of the diameter and the distance of the perpendicular from A ; that is, A B x A C = AP X AD. (Fig. 41.) Also, A B X A C = A K\ (Fig. 41.) XXXI. In a circle E D F whose centre is C, and radius C E, if tbc points B, A, be so placed in the diameter produced, that C B, C£, CA be in continual proportion, then two lines BD, A D drawn from these points to any point in the circumference of the circle will always be in the given ratio of BE to A E. (Fig. 42.) XXXII. In a circle, if a perpendicular D B be let fall from any point D, upon the diameter C I, and the tangent D O drawn from D, tken AB, AC, AO, will be in continual proportion. (Fig. 43.) XXXIII. If a triangle B D F be inscribed in a circle, and a per- pendicular D P let fall from D on the opposite side B F, and the <iiMieter D A drawn ; then, as the perpendicular is to one side inclading the angle D, so is the other side to the diameter of the ciitle; that is, D P : D B :: D F : D A. (Fig. 44.) XXXI y. The rectangle of any two sides of an inscribed triangle is equal to the rectangle of the diameter, and the perpendicular on tl»e third side ; that is, BDxDF=ADxDP. (Fig. 44.) XXXV. If a triangle B A C be inscribed in a circle, and the angle A hiflectcd by the right line A E D, then as one side is to the seg- nicDt of the bisecting line within the triangle, so is the whole bisect- ing line to the other side; that is, A B : AE :: AD : A C; and ABxAC = BE.EC + AE2. (Fig. 45.) XXXVI. If a quadrilateral A BCD be inscribed in a circle, the mn of two opposite angles is equal to two right angles; that is, ADC-l-ABC= two right angles. (Fig. 46.) [Euc. B. III. Prop. 22.] XXXVII. If a quadrangle be inscribed in a circle, the rectangle of the diagonals is equal to the sum of the rectangles of the opposite mde»; or c A X BD = CB x DA + CD x A B. (Fig. 46.) 108 CIRCLES, ETC. [pART I. XXXVIII. A circle is equal to a triangle whose base is the cir- cumference of the circle ; and height, its radius. XXXIX. The area of a circle is equal to the rectangle of half the circumference and half the diameter. XL. Circles (that is, their areas) are to one another as the squares of their diameters, or as the squares of the radii, or as the squares of the circumferences. [Euc. B. XII. Prop. 2.] XLI. Similar polygons inscribed in circles, are to one another as the circles wherein they are inscribed. XLII. A circle is to any circumscribed rectilineal figure, as the circle's periphery to the periphery of the figure. XLIII. If an equilateral triangle ABC be inscribed in a circle ; the square of the side thereof is equal to three times the square of the radius ; that is, A B* = 3 A D^. (Fig. 47.) XLIV. If from any point D in the circumference of a circle, having inscribed in it an equilateral triangle, chords be drawn to the three angles A, B, C ; the longest chord A D, is equal to the sum of the two lesser chords, B D and C D. (Fig. 48.) XLV. A square inscribed in a circle, is equal to twice the square of the radius. XLVI. The side of a regular hexagon inscribed in a circle, is equal to the radius of the circle. XLVII. If two chords in a circle mutually intersect at right angles, the sum of the squares of the segments of the chords is equal to the square of the diameter of the circle ; that is, A P^ -f P B* -r P C*^ -h P D' = diam.^. (Fig. 49.) XLVIII. If the diameter P Q be divided into two parts at any point R, and if R S be dmwn per]>endicular to P Q ; also R T ap- plied equal to the radius, and T R produced to the circumference at V: then, PR + RQ ., ., ., \ R T = ; or R T is the arithmehcal mean, ] between the f two segments RS = n/pR X RQ; or RS is the geometrical me&n. ;of the diame- « ^ « inT^ ' I ter P R, R Q RV = : or RV is the Aar mow iW mean. ^ ^^^f^' ^ PR + RQ XLIX. If the arcs P Q, QR, RS, &c., be equal, and there be drawn the chords P Q, PR, PS, &:c., then it will be P Q : P R :: PR : PQ -f PS :: PS : PR f PT :: PT : PS -f P V, &c. (Fig. 51.) L. If the arcs PQ, Q R, R S, &c. be equal, the angles Q P R, RP8, SPT,&c., will be equal ; or in equal circles, equal angles stand upon equal arcs, whether they be at the centres or circumferences. (Fig. 51.) [Euc. B. III. Prop. 20.] LI. The centre of a circle being O, and P a point in the radius, or in the radius produced ; if the circumference be divided into as many equal parts A B, B C, C D, &c., as there are units in 2 n, and lines be drawn from P to all the points of division ; then shall the continual product of all the alternate lines, viz. PAx PC x PE, &c.. CHIP. III.] PLANES AND SOLIDS. 109 be = r" — x" when P is within the circle, or = a? — r* when P is without the circle; and the product of the rest of the lines, viz., PB X P D X P F, &c., = r" -H ;i^ : where r = A O the radius, and « = 0P the distance of P from the centre. (Fig. 52.) Sect. VI. Of Planes and Solids. DEFINITIONS. 1. The common section of two planes, is the line in which they meet, or cat each other. 2. A line is perpendicular to a plane, when it is perpendicular to erenr line in that plane which meets it. 3. One plane is perpendicular to another, when every line of the one, which is perpendicular to the line of their common section, is perpendicolar to the other. 4. The inclination of one plane to another, or the angle they fonn between them, is the angle contained by two lines, drawn from anr point in the common section, and at right angles to the same, ODC of these lines in each plane. 5. Parallel planes are such as being produced ever so far in any direction, will never meet, or which are everywhere at an equal per- pendicular distance. 6. A solid an^ is that which is made by three or more plane tngles, meeting each other in the same point. 7. Similar solids^ contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same nomber of similar planes, alike placed. 8. A prism is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms. (Fig. 53 and 54.) 9. A prism takes particular names according to the figure of its hue or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c. 10. A right or upright prism^ is that which has the planes of the fldftf perpendicular to the planes of the ends or base. (Fig. 53.) When such is not the case it is called an oblique prism. (Fig. 54.) 11. A parallelapipedj or parallelopipedon, is a prism bounded by Bx parallelograms, every opposite two of which are equal, alike, and parallel. (Fig. 55.) 12. A rectangular parallelopipedon is that whose bounding planes >re all rectangles, which are perpendicular to each other. (Fig. 56,) 13. A cube is a square prism, being bounded by six equal square sdea or faces, which are perpendicular to each other. (Fig. 57.) 14. A cylinder is a round prism having circles for its ends ; and is conceived to be formed by the rotation of a right line about the cirtufflferences of two eqtuJ and parallel circles, always parallel to ^axis. (Fig. 58.) 110 SOLID GEOMETRY. [PART I. 15. The axis of a cylinder is the right line A B joining the centres of the two parallel circles, about which the figure is described. IG. A pyramid is a solid whose base is any right-lined plane figure, and its sides triangles, having all their vertices meeting together in a point above the base, called the vertex of the pyramid. (Fig. 59.) 1 7. Pyramids, like prisms, take particular names from the figure of their base. 18. A cone is a round pyramid having a circular base, and is con- ceived to be generated by the rotation of a right line about the circumference of a circle, one end of which is fixed at a point aboYe the plane of that circle. (Fig. 60.) 1 9. The axis of a cone is the right line, A B, joining the Tertex, or fixed point, and the centre of the circle about which the figure is described. 20. When the axis of a cone or pyramid is perpendicular to the base, it is called a right cone or pyramid ; but if inclined it is called oblique, 21. Similar cones and cylinders^ are such as have their altitudes and the diameters of their bases proportional. 22. A sphere is a solid bounded by one curve surface, which is everywhere equally distant from a certain point within, called the centre. It is conceived to be generated by the rotation of a semi- circle about its diameter, which remains fixed. (Fig. 61.) 23. The axis of a sphere is the right line about which the semi- circle revolves, and the centre is the same as that of the reyolving semicircle. 24. The diameter of a sphere is any right line passing through the centre, and terminated both ways by the surface. 25. The attitude of a solid is the perpendicular drawn from the vertex to the opposite side or base. PROPOSITIONS. Prop. I. If any prism be cut by a plane parallel to its base, the section will be equal and like to the base. II. If a cylinder be cut by a plane parallel to its base, the section will be a circle, equal to the base. III. All prisms and cylinders, of equal bases and altitudes, are equal to each other. [Euc. B. XI. Prop. 31.] IV. Rectangular parallclopipcdons, of equal altitudes, are to each other as their bases. [Euc. B. XI. Prop. 32.] V. Rectangular parallelopipedons, of equal bases, are to each other as their altitudes. VI. Because prisms and cylinders are as their altitudes, when their bases are equal : and, as their bases when their altitudes are equal. Therefore, universally, when neither are equal, they arc to one another as the product of their bases and altitudes : hence, also, these products are the proper numeral measures of their quantities or magnitudes. CHAP. III.] SOLID GEOMETRY. Ill VII. Similar prisms and cylinders are to each other as the cubes of their altitudes, or of any like linear dimensions. [Euc. B. XI. Prop. 33.] VIII. In any pyramid a section parallel to the base is similar to the base ; and these two planes are to each other as the squares of their distances from the vertex. IX. In a right cone, any section parallel to the base is a circle ; and this section is to the base as the squares of their distances from the vertex. X. All pyramids and cones of equal bases and altitudes, are equal to one another. XI. Every pyramid is the third part of a prism of the same base and altitude. [Euc. B. XII. Prop. 10.] XII. If a sphere be cut by a plane, the section will be a circle. XIII. Every sphere is two-thirds of its circumscribing cylinder. XIV. A cone, hemisphere, and cylinder of the same base and altitude, are to each other as the numbers 1, 2, 3. [Euc. B. XII. Prop. 10.] XV. All spheres are to each other as the cubes of their diameters; all these being like parts of their circumscribing cylinders. [Euc. B. XII. Prop. 18.] XVI. There are only three sorts of regular plane figures which can be joined together, so as to form a solid angle ; viz. three, four, or five triangles^ three squares^ and three pentagons: thus giving five different species of solid angles, which are those of the five regular or Platonic bodies ; viz. : — 1. The tetraedron^ (^^g* ^2,) bounded by four equilateral tri- angles^ each solid angle of which is formed by three triangles, 2. The kexaedron^ or cuhcy (Fig. 57,) contained by six squares^ each solid angle of which is formed by three squares. 3. The octaedron, (Fig. 63,) bounded by eight triangles^ each solid angle of which is formed by four triangles, 4. The dodecaedron^ (Fig. 64,) bounded by twelve pentagons^ each solid angle of which is formed by three pentagons, 5. The icosaedron, (Fig. 65,) bounded by twenty triangles, each solid angle of which is formed by five triangles, XVII. Only one sort of the foregoing five regular bodies, joined at their angles, can completely fill a solid space, without leaving any Tacuity ; viz. eight hexaedrons, or cubes. XVIII. A sphere is to any circumscribing solid B F, (all whose planes touch the sphere,) as the surface of the sphere to the surface of the solid, r Fig. 66.) XIX. All bo£es cii*cumscribing the same sphere, are to one an- other as their surfaces. XX. The ^here is the greatest or most capacious of all bodies of equal surface. 112 PRACTICAL GEOMETRY. [PART I. Sect. VII. Practical Geometry. It is not intended in this place to present a complete collection of Geometrical Problems, but merely a selection of the most useful, especially in reference to the employments of Mechanics and En- gineers. The instruments for the purposes of geometrical construction, are too well known to require any description here; and their use is much easier learned by an examination of the instruments themselves, than by any written explanation. Prob. I. From a given point B in a given straight line A C, to draw a line perpendicular to the same. From B as a centre, witli any radius, describe arcs cutting A C in a and c, then from a and c with any larger radius describe arcs cutting each other in d and e ; then the straight line d'Be vnW be perpendicular to A C. (Fig. 6'7.) Prob. II. To erect a perpendicular at the end of a given line. With any radius, and from a point somewhere above A B, describe a semicircular arc passing through the point B, at which the perpen- dicular is to be erected, and also cutting A B in C ; then, through C and the centre of the arc produce the right line C D, cutting the arc in D, and a line joining D and B will be the perpendicular required. (Fig. 68.) Prob. III. To bisect any given angle A C B. From C as a centre, with any radius, describe an arc cutting the sides in D and £ ; then from D and £, as centres with the same radius, describe arcs cutting each other in F ; then the straight line joining C and F will bisect the angle A C B. (Fig. 69.) Prob. IV. To bisect a given angle BAP; then to bisect its half; and so on. Through any point B draw B E parallel to A P, and upon B E set off the distance B C equal to B A ; then join A C^ and it will bisect the angle BAP. Again, set off, upon B E, from C, C D = C A ; join A D, and it will bisect C A P, or quadrisect BAP. Again, set off, upon B £, D E = D A ; join £ A ; and E A P will be ^ of B A P : and so on. (Fig. 70.) Prob. Y. At a given point A in a given line A B, to make an angle equal to a given angle C. From the centres A and C, with the same radius, describe the arcs D E, F G, Then, with radius D £, and centre F, describe an arc, cut- ting F G in O. Through G draw the line A G ; and it will form the angle required, (Fig. 71 .) Prob. VI. To divide any given angle ABC into three equal parte. From B, with any radius, describe the circle A C D A. Bisect the angle A B C by B £, and produce A B to D. On the edge of a ruler CHAP. III.] PRACTICAL GEOMETRY. 113 mark off the length of the radius A B. Lay the ruler on D, and move it till one of the marks on the edge intersects B £, and the other the arc A C in O. Set off the distance C G from G to F : and draw the lines B F, and B O, they will trisect the angle ABC. (Fig. 72.) Prob. VII. To divide a given line A B into any proposed number of equal parU, Ist Method. Draw any other line AC, forming any angle with the given line A B ; on which set off as many of any equal parts, A D, D E, E F, F C, as the line A B is to he divided into. Join B C ; parallel to which draw the other lines F G, EH, D I : then these will divide AB in the manner required. (Fig. 73.) 2nd Method^ without drawing parallel lines. Let A B he the line which is to be divided into n equal parts. Through one extremity A draw any right line A D, upon which set off n -f 1 equal parts, the point D being at the termination of the (n + l)th part. Join DB and produce it until the prolongation BE = B D. Let F be the termination of the (n — l)th part. Join FE, and the right line of junction will cut the given line AB in the point P, such that B P = - A B ; and of course n distances each equal to B P set off upon B A, will divide it, as required*. (Fig. 74.) Pbob. VIII. To cut off from a given line A B, supposed to he tery shorty any proportional part. Suppose, for example, it were required to find the -y^, -^^^ j\r, &c. of the line A B, fig. 75. From the ends A and B draw AD, B C, perpendicular to A B, and divide A D into twelve equal parts ; then through these divisions 1, 2, 3, &c., draw lines \f 2^, &c., parallel to A B. Draw the diagonal A C, and 1 d will be the -^-^ of A B ; 2 c, -^j, and so on. The same method is applicable to any other part of a given line. Prob. IX. To make a diagonal scale^ say^ offeety inches, and tenths of an inch. Draw an indefinite line A B, on which set off from A to B the given length for one foot, any required number of times ; and from these divisions A, C, H, B, draw AD, CE, &c., perpendicular to AB. On A D and B F set off any length ten times, and through these divisions draw lines parallel to A B ; then divide A C and D E into twelve equal parts, each of which will be one inch. Draw the lines A 1, ^2, &c., and they will form the scale required; viz., each of the larger divisions from E to G, G to F, &c., will represent a foot ; each of the twelve divisions between D and £, an inch ; and the * The truth of thie method is easily demonstrated. Through i the inter- mediate point of division, on a n, between f and d, draw i b. Then, because D B « B K, and D I B 1 F, I B is parallel to F P. Consequently, b P : B a : : I P : I A : : 1 : n, by construction. I 114 PRACTICAL OBOMBTBY. (^r^^. . several horizontal lines parallel to R C in the triangle £ C R, will be equal to -j\^, -f%, -,»ff, &c., of an inch. (Fig. 76.) Note. — If the scale be meant to represent feet, or any other onit, and tenths and hiindredfks^ then D E must be divided into ten instead of twelve equal parts. Prob. X. To fitid the centre of a circle. Draw any chord A B, and bisect it perpendicularly with the line R D. Then bisect R D in C, which will be the centre required. (Fig. 29.) Prob. XI. To divide a given cirde into any number ofeoneentrk parts^ equal to each other in area. Draw the radius A B, and on it describe the semicircle Afe dB. Divide AB into the proposed number of equal parts, 1, 2^ 3, &c., and erect the perpendiculars 1 rf, 2e, Sf &c., meeting the semicircle in rf, e,yj &c. Then from the centre B, and ^nth radii Be/, Be, &c. describe circles ; so shall the given circle be divided into the proposed number of equal concentric parts. (Fig. 77.) Prob. XII. To divide a given circle into any number qf partij equal both in area and perimeter. Divide the diameter Q R into the proposed number of equal parts at the points S, T, V, &c. ; then, on one side of the diameter describe semicircles on the diameters QS, QT, QV, and on the other side of it describe semicircles on RV, RT, RS; so shall the parts 17, 35, 53, 71, be all equal> both in area and perimeter. (Fig. 78.) Prob. XIII. To describe the circumference of a circle through three given points^ A, B, C. From the middle point B draw chords B A, B C, to the two other points, and bisect these chords perpendicularly by lines meeting in O, which will be the centre. Then from the centre O, at the distance of any of the points, as O A, describe a circle, and it will pass through the two other points B, C, as required. (Fig. 79.) The same method may be employed for finding the centre of a circular arc, by taking any three points in the same, as A, B, and C. Prob. XIV. To describe mechanically the circumference of a cireU through three given points^ A, B,C, tchen the centre is inaccessible^ or the circle too large to be described with compasses. Place two rulers M N, R S, cross ways, touching the three points ABC. Fix them in V by a pin, and by a tmasverse piece T. Hold a pencil in A, and describe the arc B A C, by moving the angle RAN so as to keep the outside edges of the rulers against the pins B C. Remove the instrument R V N, and on the arc described mark two CHAP. III.] PRACTICAL GEOMSTBY. lid points, D, E, 80 that their distance Bhall be equal to the length B C. Apply the edges of the instrument against D £, and with a pencil in 6 describe the arc B C, which will complete the circumference of the circle required. (Fig. 80.) Otherwise, — Let an axle of 12 or 15 inches long carry two unequal wheels A and B, of which one, A, shall be fixed, while the other, B, shall be susceptible of motion along the axle, and being placed at any assigned distance, A B, upon the paper or plane on which the circle is to be described. Then will A and B be analogous to the ends of a conic frustum, the vertex of the complete cone being the centre (O) of the circle (C D E F) which will be described by the rim, or edge, of the wheel A, as it rolls upon the proposed plane. Then it will be, as the diameter of the wheel A is to the difference of the diameters of A and B, so is the radius of the circle proposed to be described by A, to the distance, A B, at which the two wheels must be asunder, measured upon the plane on which the circle is to be described. The wheel B will evidently describe, simultaneously, another circle (O H I K) whose radius will be less than that of the former bv A B. (Fig. 81.) Pbob. XV. On a ^ven chord ABto describe mechanicaUy an arc of a large circle that shall contain any number of degrees. Place two rulers, forming an angle A C B, equal to the supplement of half the given number of degrees, and fix them in C* Place two pins at the extremities of the given chord, and hold a pencil in C ; then move the edges of this instrument against the pins, and the pencil will describe the arc required. (Fig. 82.) Suppose it is required to describe an arc of 50 degrees on the given chord A B ; subtract 25 degrees (which is half the given angle) from 180, and the difference, 155 degrees, will be the supplement. Then form an angle A C B of 155 degrees with the two rulers, and proceed «s has been shown above. Pbob. XVI. To describe the segment of a circle of large radius^ of which the chord A B and versine C D are given. Through D draw H I parallel to A B ; then join A D and D B, and draw A H perpendicular to A D and B I perpendicular to B D, also draw A d and B d perpendicular to A B. Then divide A C, B C, H D, and I D, into any number of equal parts, 1, 2, 8, &c., and draw lines joining the corresponding numoers in H I and A B ; also divide A c/, B d, into the same number of equal parts, a, by c^ &c., and draw lines from the point D to these last divisions, then will the points where the lines a D, 6 D, c D, &c. cut the lines 1 1, 22, 33, &c., be so many points in the required segment. (Fig. 83.) Pbob. XVII. To find the length of any given arc of a circle^ A B. From A and B, as centres, widi radius equal to a quarter of the diord of the arc A B, describe arcs cutting the given arc in C, and its diord ID D ; join C D, which will equal h^f the length of the aro A B nearly. (Fig. 84.) I 2 116 PRACTICAL GEOMETRY. [PART I. Prob. XVIII. To draw a straight line equal in length to any given portion of the circumference of a circle. Let A B C D be the circle, and let A a, a ^, 6 B, be the portions of the circumference of wliich the length is required ; draw^ the two diameters B D and A C at right angles to each other, and at the end of the latter draw the tangent line Ae ; then divide the radius EC into four equal parts, and set off three of them from C to F on the diameter produced ; then draw lines from F through the points a,d,B, cutting the tangent line A^, in c, rf, e; so shall the lines Ac, c</, de, be equal in length to the arcs A a, a 6, 3 B, and the whole line Ae equal to the quadrant A B. (Fig. 85.) Prob. XIX. To bisect any given triangle ABC. Upon any one of the sides, as A B, describe a semicircle, which bisect in the point D ; then from B as a centre with radius B D de- scribe an arc cutting A B in E, through which point draw the line E F parallel with the side A C ; then will the line £ F bisect the given triangle ABC. (Fig. 86.) Prob. XX. To reduce a given rectilinear figure, ABCDEFOA, of any number of aides, to a triangle of equal area. Join any two alternate angles, as A C, and through the interme- diate angle B, draw B H parallel to A C, cutting one of the adjoining sides in H, and join C H, then will the triangle C B a, added to the figure, be equal to the triangle a A H taken out of the same ; in like manner join H D, and draw G I parallel to the same through the in- termediate point O, producing it to cut the side A G (also produced) in I, and join I D. Next join D F, and through E draw a line paral- lel thereto, cutting the side G F produced in K, and join D K ; lastly, join D G, and draw parallel to the same through the point K a line cutting the side A G produced in L, then join D L, and the tri- angle I D L will be equal in area to the given rectilinear figure ABCDEFGA. (Fig. 87.) Prob. XXI. To form a rectangle of the largest area, in a given triangle, ABC. Bisect any two sides A B and B C in D and E, and from those points draw lines D F and £ G perpendicular to the third side A C, and join D E, then will the rectanele D E F G be the largest which can be inscribed in the given triangle. (Fig. 88.) Prob. XXII. To form a square equal in area to a given triangley ABC. On the longest side produced set off C D, equal half the perpendi- cular height B E ; and at C erect the perpendicular C F ; then on A D describe a semicircle cutting C F in G, and on Q G form the square CGHI, which will be equal in area to the given triangle. (Fig. 89.) CHAP. III.] PRACTICAL OEOIIBTRY. 117 Pbob. XXIII. To find the 9ide of a square eqtuU in area to a given rectangle^ A B C D. Produce the lesser side A B of the rectangle till A E equal the longer side AD; then describe upon A £ a semicircle cutting B C in F, and join A F, which will be the side of the square required. (Fig. 90.) Pbob. XXIV. To find the side of a square egtmlto the difference beiveen two given squares. Let the sides of the two given squares be A and B. Then draw the line C D equal to the lesser line B, and at one of its extremities erect the indefinite perpendicular D £ ; then with radius equal to A, and from C as a centre, describe an arc cutting this perpen- dicular in E ; then D E will equal the side of the square required. (Fig. 91.) Pbob. XXV. To find the side of a square equal in area to any number of given squares. Let the lines A, B and C be the sides of the given squares. Draw DE equal to A, and at the end of it erect the perpendicular E F equal to B ; join D F and perpendicular to it, from the point F, erect the perpendicular O F equal to C ; then join D O, which will be equal to the side of a square equal in area to the three squares on A, B, and C. (Fig. 92.) Pbob. XXVI. To find the side of a square nearly equal in area to a given circle, A B C D. Draw the two diameters, AC, B D, at right angles to each other ; then bisect the radius E C in F, and through D and F produce the straight line D G cutting the circle in G, then will D G equal the side of the square required. (Fig. 93.) Pbob. XXVII. Given the side of a regular polygon of any number of tides (not exceeding twelve% to find the radius of the circle in which it may be inscribed. Multiply the given side of the polygon by the number which stands in column 6, opposite its proper name in the annexed Table of Polygons ; the product will be the radius required. Thus, suppose the polygon was to be an octagon, and each side 1 2, then 1-3065628 X 12 = 15-6687536 would be the radius sought. Take 15*67 as a radius from a diagonal scale, describe a circle, and from the same scale, taking off 12, it may be applied as the side of an octagon in that circle. Pbob. XXVIII. Given the radius of a circle, to find the side of any regular polygon (not having more than twelve sides) inscribed in it. Multiply the given radius by the number in column c, standing opposite the number of sides of the proposed polygon : the product is the length of the side required. Thus, suppose the radius of the circle to be 5, then 5 x 1*732051 as 8*66025, will be the side of the inscribed equilateral triangle. 118 PKAOTICAL OSOMITBT. TABLE OP POLYGONa [Fi 6 "5 Multipliers Radius of Fac Names. for areas. circmD. cir. fori Z,'^ («) (ft) ( 8 Trigon 0-4330127 0-5773503 1-73S 4 Tetragon, or Square 1-0000000 0-7071068 1-41^ 5 Pentagon 1-7204774 0-8506508 117^ 6 Hexagon 2-5980762 10000000 1-00( 7 Heptagon 3-6339124' 1- 1523824 0-86' 8 Octagon 4-8284271 1-3065628 0-76, 9 Nonagon 6-1818242 1-4619022 0-68- 10 Decagon 7-6942088 1-6180340 0-61J 11 Undecagon 9-8656399 1-7747324 0-56; 12 Dodecagon 111961524 1-9318517 0-51' Pros. XXIX. To reduce a simple rectilinear figure to a i one upon either a smaller or a larger scale. Pitdi upon a point P any where about the given figure A B either within it, or without it, or in one side or angle; but m middle is best. From that point P draw lines through all the i upon one of which take P a to P A in the proposed proportion scales, or linear dimensions ; then draw a b parallel to A B, B C, &c. ; so shall abode be the reduced figure sought, greater or smaller than the original. (Hutfons Mens.) (Fig. S Otherwise to Reduce a Figure by a Scale, — Measure all the and diagonals of the figure, as A B C D E, by a scale ; and lay the same measures respectively from another scale, in the pro] required. To Reduce a Map^ Design^ or Figure^ by Squares. — Divide tl ginal into a number of little squares, and divide a fresh paper, dimensions required, into the same number of other squares, greater or smaller, in the proportion required. Then, in every of the second figure, draw what is found in the corresponding of the first or original figure. The cross lines forming these squares may be drawn with a and rubbed out again after the work is finished. But a more and convenient way, especially when such reductions are wanted, would be to keep always at hand frames of squares made, of several sizes ; for, by only just laying them down up papers, the corresponding parts may be readily copied. These : may be made of four stifiT or inflexible bars, strung across with hairs, or fine catgut. When figures are rather complex, the reduction to a differen will be best accomplished by means of such an instrument as I sor Wallace's Eidograph^ or by means of a Pantograph^ an i ment which is now considerably improved by simply changic place of the fulcrum. See the Mechanics' Orade^ Part II. p. ; CfliP.IV.J WS10HT8 AND IfEASURKS. 119 CHAP. IV. MENSURATION. Sect. I. Weights and Measures, MmuEATiON is the application of arithmetic to geometry, by wliicb m ire enabled to discover the magnitude and dimeneious of tnj geometrical figures, whether solid or superficial. To enable us to expitis this magnitude in determinate terms, it is necessary to womeflome magnitude of the same kind as the unit, and tlien, by itatiDg how many times the given magnitude contains that unit, we obtain its measure. The different species of magnitude which have most frequently to be determined, are distinguishable into six kinds, viz. : — 1. Length. 2. Surface. 3. Solidity, or Capacity. *. force of Gravity, commonly called Weight. 5. Angles. 6. Time. The several units assumed as the standards of measurement of each of then particular species of magnitude, are entirely arbitrary, and, ttoie^ently, vary among different nations. In this kingdom they w heea fixed by Act of Parliament*, and are as follows, viz. : — Length is a yard. . Surface is a square yard, the ^g^^^ th of an acre. I (Solidity is a cubic yard. Tie standard of < i^^P???^ '' "" ^^l'"''' mmuuMuix "» \ height IS a pound. I Angular measurement is a degree, the d60th part of the circumference of the circle. Time is a day. ne values of the whole of the above are determined directly or Wifectly by comparison with the length of a pendulum, which in ^ latitude of London, placed at the level of the sea, and in a * 5 Gso. IV. c. 74 ; which took effect Itt January, 1826. 120 WBIOHTS AND 1IBA8URBS. [r^^. yacuuTD, would vibrate seconds of mean time. The length of such a pendulum being, to the length of the standard yard, as 39*1393 inches are to 36. Since the passing of this act, however, some very elaborate and scientific experimenis of Mr. Francis Daily have shown that errors of sufficient moment to be taken into the account in an inquiry of this kind, render the above proportion inaccurate*. We do not, in fact, yet know the length of a seconds* pendulum at London, vibrat- ing in the circumstances proposed. The following standard yards, made with great accuracy, give the annexed results : — Inches. General Lambton's scale, used in India ... 35*99934 Sir George Shuckburgh's scale 35*99998 General Ray's scale 36*00088 Royal Society's standard 3600135 Ramsden'sbar 36002*9 Its copy, at Marischal College, Aberdeen... 36*00244 1. MEASURES OF LENGTH. Inchcft. Feet. 12 = 1 Yard.. o/» o __ I Rod* <5t> — d — 1 orPoie*. 198 = 16J = 5J == 1 Furicmg.. 7920 = 660 = 220 = 40 = 1 Mile. 63360 = 5280 = 1760 = 320 = 8 = 1 The mean length of a degree of latitude measured on the terrestrial meridian, is 69*0444 imperial miles; the 60th part of which, or 6075*6 feet, is the length of a nautical or geographical mile, three of which are equal to a league. And the length of a degree of lo/igi- tude, measured upon the equator, is 69*1555 imperial miles. An inch is the smallest lineal measure to which a name is nven ; but subdivisions are used for many purposes. A mong mechanics the inch is commonly divided into eighths. By the officers of the revenue, and by scientific persons, it is divided into tenths.^ hundredthn^ &c. Formerly it was made to consist of 12 parts, called lines^ but these have properly fallen into disuse. Particular Measures of Length. A Nail = 2\ Inches \ Y A — • 4 O f i ^^^ ^^^ measuring cloth of all kinds Ell = 5 Quarters ) Hand = 4 Inches Used for the height of horses. Fathom = 6 Feet Used in measuring depths. Link = 7 Inc., 92 \ Used in Land Measure, to faciliti hdths. } computation of the content, Chain =100 Links ) square chains being equal to an ac * See foot-note, p. 223. f CRIP. IV.J WBIOHTS AND MKA8UBES. 121 2. MEASURES OF SURFACES. SqoRliKfaa. 144 = 1296 = 39204 = 1568160 = Sq.Feet. 1 9 = 272i = 10890 = Sq. Yards, 1 orRodi. 30| = 1 Rood.. J210 = 40= 1 a™. 6272640 = 43560 = 4840 = 160= 4= 1 mL 4014489600 = S ^878400 = 3097600 =102400 = 2560 = 640 = 1 d. HBASUBE8 OF SOLIDITY AND CAPACITY. Division I. Measures 0/ Solidity, Cubic Inchet. Cubic Feet. 1728 = 1 Cubic Yard. 46656 = 27 = 1 Division II, Mntures of Capacity for aU liquids^ and for aU dry goods^ except wc^oiare comprised in the third division, Tbe imperial gallon (the standard for all measures of capacity) contains 10 pounds imperial avoirdupois weight of distilled water, 'wghedin air at 62^ Fahr. (the barometer being at 30 inches) ; con- 'wiuently, its capacity is 277*274 imperial cubic inches. 4 Gills = 1 Pint = 34-659 Cubic Inches. 2Rnt8 = 1 Quart = 69 318 „ „ 4 Quarts = 1 Gallon = 277*274 „ „ 2 Gall. =1 Peck = 554548 „ „ 8 Gall. = 1 Bushel = 2218-192 „ „ 8 Bush. = 1 Quarter = 10-269 Cubic Feet. 5Qr8. = 1 Load = 51*345 „ „ The four last denominations are used for dry goods only. For liquids leveral denominations have been heretofore adopted, viz. : — ^or beer, the firkin of 9 gallons, the kilderkin of 18, the barrel 0(36, the hogshead of 54, and the butt of 108 gallons. These will probably con tin ae to be used in practice. For wine and "pvits, there are, the anker, runlet, tierce, hogshead, puncheon, PpC) butt, and tun; but these may be considered rather as the i^Anies of the casks in which such commodities are imported, than •• expressing any definite number of gallons. It is the practice to 8>oge all such vessels, and to charge them according to their actual cwitcnt. Flour is sold, nominally, by measure, but actually by weight, '^Itoned at 7 lb. aToirdupois to a gallon. 122 WBI0HT8 AND IfEASURIS. [PABT I, Division HI. Imperial Measures of Capacity for coalsy 'culm^ lime, fish^ potaioet^ fruity and other goods, commonly sold by hbapbd measukb : 2 Gallons = 1 Peck 8 Oallons = 1 Bushel 3 Bushels = 1 Sock 12 Sacks == 1 Chald. 28154 I ^^^^^ Inches, nearly, -gt I Cubic Feet, nearly. The goods are to be heaped up in the form of a cone, to a height above the rim of the measure of at least f of its depth. The out- side diameter of measures used for heaped goods are to be at least double the depth ; consequently not less than the following dimen- 19i inches. Bushel Half-bushel 15| inches. ! Peck 12 J inches. Gallon 9| inches. Half-gallon 7f Inches. The imperial measures described in the second and third divisiont were established by Act 5 Geo. IV. c. 74. Before that time there were four different measures of capacity used in England, yiz. : — 2. For malt liquors, the gallon of which con- ) ooo tained j ^^^ " 3. For com and all other dry goods, not | aoo.o heaped, the gallon of which contained ... ) " " 4. For coals, which did not differ sensibly from the imperial measure. Hence, with respect to wine, ale, and com, it will be expedient to possess a Table of Factors, For converting old measures into new, and the contrary. By Decimals. By Vulgar Fractions nearly. Com Measure. Wine Measure. Ale Measure. Corn Mea- sure. Wine Mea- sure. Ale Mea- sure. To convert old ) .96943 measures to new. S -83311 1-01704 n- f n To convert new J ^ 3^ 3 measures to old. S 1-20032 -98324 i\ i ^ N.B. For reducing the prices, these numbers must all be revenad. GHAP. IV.] WUOHTS AND IfSAtUBBS. 123 4. MBA8URE8 OF WEIGHT. Division L Avoirdupois Weight. The standard of weight is the avoirdupois pound, the value of 'vhich is determined by its heing the weight of 27'7274 cuhic inches ^f distilled water, weighed in air at 62"^ Fahr., with the harometer at SO inches. Avoirdupois weight is used in almost all commercial transactions, ^md in the common dealings of life; its divisions are as follows : — Dndum. Ounees. 16 s: 1 Pounds. 256 S5 16 =c 1 QiuDten. 7168 = 448 =s 28 1 Hundred- weighu. 28672 = 1792 = 112 = 4 5= 1 Ton. 573440 z^ 35840 = 2240 = 80 = 20 Particular weighu belonging to this Division. cwt. qr. lb. 8 Pounds = 1 Stone z= 8 Used for meat. 1* .> = 1 « s= 14 \ 2 Stone s= 1 Tod s= 1 e|Tod = 1 Wev = 1 2 14 I) Used in the wool trade. 2 Wejs =s 1 Sack == 3 1 12 Sacks 2= 1 Last =5 30 Division IL Troy Weight, For weighing gold, silver, and precious stones (except diamonds), troj weight is employed, its divisions are — Penny- OtaiBi (gn.) i*«ighU (dwts.) 24 =s 1 OUBCM(OI.) 480 5= 20 = 1 Pound (lb.) 5760 = 240 = 12 = 1 But troy weight is also used hv apothecaries in compounding medi- cines; and 18 then divided as follows, viz. : — Onant(gr.) Seruplo (3.) 20 = 1 Drachms (3.) 60 480 5760 i 3 = 24 = 288 ^ 1 8 96 Ounces (3 ) = 1 = 12 Pound (lb.) = 1 The troy pound is equal to the weight of 22*8157 cuhic inches of distilled water, weighed in air at 62^ Fahr., and the harometer at 30 inches; and is, therefore, less than the avoirdupois pound, in the proportion of 144 to 175. 124 WEIGHTS AND MEASURES. [PABT I. OS. dwts. gn. 1 lb. avoirdupois = 14 11 15i troy = 7000 troy grains. 1 oz. „ == 18 5$ „ = 437-5 „ 1 dr. „ =01 3| „ = 27-343 „ 1 trov lb. = 0-822857 avoirdupois lb. 1 avoir, lb. = 1-215271 troy lb. For scientific purposes the grain only is used ; and sets of weights are constructed in decimal progression, from 10,000 grs. downwards to i^J^th of a grain. The carat^ used for weighing diamonds, is 3^ grains. The term, however, when used to express the fineness of gold, has a relative meaning only. Every mass of alloyed gold is supposed to be divided into 24 equal parts; thus the standard for coin is 22 carats fine, that is, it consists of 22 parts of pure gold, and 2 parts of alloy. What is called the new standard^ used for watch-cases, &c. is 18 carats fine. 5. Angular Measure; or. Divisions of the Circle. 60 Seconds " =1 Minute, denoted by ' 60 Minutes = 1 Degree, „ ° 30 Degrees = 1 Sign „ ' 90 Degrees = 1 Quadrant. 360 Degrees, or 12 Signs = 1 Circumference. Formerly the subdivisions were carried on by sixties ; thus, the se- cond was divided into 60 thirds, the third into 60 fourths, &c. At present, the second is more generally divided decimally into lOths, lOOths, &c. The degree is frequently so divided. 6. Measure of Time. 60 Seconds = 1 Minute. 60 Minutes = 1 Hour. 24 Hours = 1 Mean Solar Day. 23 H. 56 M. 3-5 S. = 1 Sidereal Day. 7 Days = 1 Week. 28 Days = 1 Lunar Month. 28, 29, 30, or 31 Days = 1 Calendar Month. 12 Calendar Months = 1 Year. 365 Days = 1 Common Year. 366 Days = 1 Leap Year. 365jDavs = 1 Julian Year. 365 D. 5 H. 48 M. 45| S. = 1 Solar Year. 365 D. 6 H. 9 M. 11 S. =1 Sidereal Year. A solar day is the time that elapses between two successive transits of the sun over the same meridian, and is not always of equal dura- tion, being longer at some seasons of the year than at others ; the difference between the actual length of a solar day and the mean length, is called the equation of time. A sidereal day is the interval of time that elapses between two successive transits of any fixed star CHiP, IV.] WEIGHTS AND MRA8UBBS. 125 OTer the same meridian, and is tbe most uniform of all astronomical pen'ods, neither theory nor observation having detected the slightest TSjiatioD in its length. A solar year is the time in which the earth passes through the twelve signs of the zodiac, and is the natural year, Dccause it always keeps the same seasons in the same months. The wd^eal year is the time that elapses between the eartb leaving any fised star and returning to it again. In 400 years, 97 are leap years, and 303 common. Tbe same remark, as in the case of angular measure, applies to tbe mode of subdividing the second of time. COMPARISON OF ENGLISH AND FRENCH WEIGHTS AND MEASURES. The following is a comparative Table of the Weights and Measures ^i England and France, which were published by tbe Royal and dJoitral Society of Agriculture of Paris, in the Annuary for 1829, ^uid founded on a Report, made by Mr. Mathieu, to the Royal Aca- demy of Sciences of France, on the bill passed the 17th of May, 1824, ^■^latiTe to the Weights and Measures termed " Imperial," which are xaow used m Ghreat Britain. ENGLISH. 1 Inch (l-36th of a yard) 1 Foot (l-3rd of a yard 'Yard imperial Fathom (2 yards) . ^ole, or perch (5^ yards Fnrlong (220 yards) Mile (1760 yards) FRENCH. 1 Mill; 1 Centimetre 1 Decimetre . IMetre . ^yriamctre , Measures of Length. FRENCH. 2*539954 centimetres. 3-0479449 decimetres. 0-91438348 metre. 1-82876696 metre, 5-02911 metres. 201-16437 metres. 1609*3149 metres. ENGLISH. 0-03937 inch. 0-393708 inch. 3-937079 inches. 39-37079 incbes. 3-2808992 feet. 1-093633 yard. 6*2138 miles. Square Measures. ENGLISH. J Yard sqoare Jod (square perch) 1 Bood (1210 yards square) ' Acre (4840 yards square) FRENCH. J Metre square \^ IHeetare . . FRENCH. 0-836097 metre square. 25*291939 metres square. 10-116775 ares. 0-404671 hectare. ENGLISH. 1-196033 yard square. 098845 rood. 2-473614 acres. 126 WnOHTS AND ICEABOBKt. [pAWr I Solid Measures, ENGLISH. 1 Pint (l-8th of a gallon) 1 Quart (l-4th of a gallon) 1 Gallon imperial . 1 Peck (2 gallons) 1 Bushel (8 gallons) 1 Sack (3 bushels) 1 Quarter (8 bushels) 1 Chaldron (12 sacks) FRENCH. 1 Litre . . = . I Decalitre . . . . = . 1 Hectolitre . . . = , Weights. ENbl.ISH TROY. 1 Grain (l-24th of a pennyweight) = 1 Pennyweight (1 -20th of an ounce) = 1 Ounce (1-1 2th of a pound troy) =: 1 Pound troy imperial . = ENGLISH AVOIRDUPOIS. 1 Drachm (l-lOth of an ounce) . = 1 Ounce (l-16th of a pound) = 1 Pound avoirdupois imperial = 1 Hundred- weight (112 pounds) = 1 Ton (20 hundred-weight) . = FRENCH. FRENCH. 0-567932 litre. M35864 litre. 4-54345794 litres. 908G9I59 litres. 36-347CG4 litres, 1-09043 hectolitre. 2-907813 hectolitres. 1308516 hectolitres. ENGLISH. 1-760773 pint 0-2200967 gallon. 2-2009667 gallons. 22-009667 gallons. FRENCH. 0*06477 gramme. 1*55456 gramme. . 31-0913 grammes. 0-3730959 kilogramme. FRENCH. 1-7712 gramme. . 28-3384 grammes. 0-4534148 kilogramm*. . 50-78246 kilogiammm. 1015-649 kilogrammes. ENGLISH. 1 Gramme Kilogramme 15-438 grains troy. 0-643 pennyweight. 0032 16 ounce troy. 2-68027 pounds troy. 2*20548 pounds avoirdupoil Angular Measure, In France, the centesimal division of the circle is frequently em ployed, in which the whole circumference is divided into 400 degre« each degree into 100 minutes, each minute into 100 seconds, &c. CENTESIMAL. ENGLISH. 1 Degree . . . , = . 54 minutes. 1 Minute . . . = . 32*4 seconds. 1 Second . . . . = . *324 second. ENGLISH. 1 Degree 1 Minute 1 Second- CBNTESIMAL. 1^ degree. 1-85185 minute. 8*08641 seconds. CHAP. IV.] MBOTUJEUTION OF SUPBRFICIE8. 12? Sect. II, Mensuration of Superficies* TRIANGLES. Let by c, and dy represent the three sides of a triangle (see fig- 95); 0j y, and ^, the angles opposite those sides respectively; and h the perpendicular height from the vertex to the hase b; then the area = | & A, or = I ^ c . sin ^ = ^ c (/ . sin jS = ^ </& . sin 7. Or, = a/ </^ — f — — H - ) • 2 where b is the greatest side and c the least If half the sum of the three sides or = », then the area = V» {s — A) . (a — c) . (« — d)y and the log of area =i{log« 4- log (« — i) -f log(« — c) -|- log (« — «?)}. For the method of obtaining the unknown sides or angles of triangles from those which are known, see Chap. V., Sect. IV. QUADRILATERALS, OR FOUR-SIDED FIGURES. Square^ Rectangle^ Rhombus and Rhomboid: — To obtain the area, naltiplj the perpendicular height (^, fig. 96) by the base {b) on which it falls. The area of a Trapezium is best found by dividing it into two triangles (as fig. 17), the areas of which may be found by the foregoing rules. The area of a Trapezoid (fig. 16) may either be found in the same way, or by multiplying half the sum of the two parallel sides (a -f b) by the perpendicular distance between them (Ji). POLYGONS. To obtain the area of any regular polygon, having less than twelve sides, multiply the square of one of the sides by the number found in colamn a, of the Table of Polygons, page 118. Or, generally, if / =: the length of one of the equal sides, and n the number of them; then the area = /^ •- tan 4 ^90 n— ISOX The area of an irregular polygon may be determined by dividing it into triangles, as in fig. 97. Or by forming one triangle equal in area to it, by Prob. XX., page 116. CIRCLES. Let d represent the diameter, e the circumference, a the area, mnd/» := 3*14159 (see Table V. in the Appendix) ; then the 128 IfBNSURATION OF 8UPBRFICIB8. [PART I. _. , c 4a ^ /a Diameter =:(/ = - = — =2 a/~« p c ^ p 4a Circumference = c =/? {/ = — = 2 ^p a. Area = « ='^,^ = /- = ^' = 'TSS*./'. 4 4/? 4 ^ "" 5 "~ rf* "~ 4a' Circular Arcs. — If r represents the radius, and d the diameter of the circle; a the sine, and r, the versed sine of the arc; c the chord of half the arc, v^ the versed sine of half the arc; and m the mea- sure in degrees of the whole arc ; then the length of the arc = -0174533 rm; °'=2rf^^^^^ nearly; 8c — 2« or = nearly. And the following relations between the several quantities, firom which any one of them may be obtained, will be found useful, viz.: — c, =r-N/i^^=7 .' (1.) c=n/ ^^H-^^ (2.) «= N/r'^-(r-t?J* (3.) ^=f + t^x (*.) Circular Sectors. — Let d represent the diameter of the circle, / the length of the arc of the sector^ and m its measure in degrees; then the area = | r /. or = -00218 J^w. Circular Segments, —Let d represent the diameter of the cirdc, © the versed sine, c, the chord of the whole arc, and c^ the chord of half the arc; then the area = | {-^{dv — ©^j ^ ^ ^dv} nearly; or = 1 1? (</«? — 1^ c^) nearly; <>r =^a<^(<^i +T<^2) nearly; or = -JtJ N/(i<?i^ + iv^) nearly. * Table VI. in the Appendix contains the diameter, circumference, area, and length of the side of an equal square to circles from 1 to 100*76 in diameter. CHIP. IV. j MENSUBATION OF 8UPBBFICIE8. 1 29 Or the area of any segment, as fig. 98, may be obtained by finding (lie area of the sector A BCD, and subtracting from it tbe area of the triangle A C D. Tbe area of a circular zone^ as £ F G H, may be found by subtract- ing the area of the segment GH I from tbe area of tbe segment E6IHF. PABA30LA. The area of a parabola is equal to two-tbirds of the product of tbe l«e X the perpendicular height. For the relations between die abscissee and ordinates in tbe para- bola, and the method of deriving one from the other, see Chap. VI. page 172. Psrahdic Arcs, — Let x be the abscissa a 5, ^g, 99, measured from the vertex a, and y the corresponding right ordinate cb; then the length of the btc cadis = ^y/(f-ri^) nearly; = |{V(/ -h*^)-^ 4 y Parabolic Frustunij or Zone, — The word zone is here used to clcDotethe space ede^ (fig. 99), contained between the two parallel doable ordinates cd and e^. Let ^j = the length of the double ^'fliMte ed^ and y^ = the length of e^y and d = the perpendicular datance between them bf; then the S ..3 area of the zone = 4 ^ ^-^, ^^ • ELLIPSE. The area is equal to the product of the transverse diameter x the wnjngate diameter x 785398. '^periphery or circumference is equal to the sum of the trans- 'eweand conjugate diameters x 1*57079. The foregoing is only an approximation to the periphery, although wfidently near the truth for ordinary purposes ; but where greater ■ewracy is required the following series may be employed. Let t '^P'ttcnt the transverse axis, c the conjugate, p = 3* 14159, &c., «d <^= 1 — -; then r ^j, d 8d' SK5d^ 3\5\7d* win be the periphery. EBipOe Segments, — To find tbe area of tbe elliptic segment abc (6g. 100), find tbe area of the corresponding circular segment, dbe^ Meribcd on the same axis (bf) to which tbe cutting line or base of K 130 IfBNSUBATION OF 80LID8. [PAB the segment ae is perpendicular. Then, as this axis (hf) : other axis (ph) :: the circular segment (dbe) : the elliptic ment (a be). Elliptic Arcs. — Let t represent the semi-transverse, and e semi-conjugate diameters of the ellipse, and d the distance of ordinate from the centre, then the length of the arc conta hetwecn the ordinate and the parallel semi-axis will be -'^'''e?^'^-iO^'''' 112?^ ^ + *^ f^ — c^ or make ^ — = r ; then the length of the arc = €? ^ / —5 — . ^ nearly. HYPERBOLA. The area of an hyperbola or hyperbolic segment may be founc follows : — let t and c represent the semi-transverse and semi-coi gate diameters, 2y the double ordinate which cuts off the segmt and a its abcissa ; also g = : then the area = 2:»yfi -1 ?1 ^ &c} ^^^ 3.5 3.5.7 3.5.7.9 ^ or =-£^(4 ^(2tx 4. f ar^) 4- ^Ztx] nearly. Hyperbolic Arcs. — The notation being the same, the follow approximation may be employed to obtain the length of an hy] bolic arc. 120c-^ + (19/* + 21c')4« ^'^ = 120c^/-K9f^ + 21c*)4^ +y' ^«^^y- Sect. III. Mensuration of Solids. PARALLELOPIPEDON, PRISM, OR CYLINDER. Surface. — Multiply the perimeter of one end by the lenstl: perpendicular height, to which add the area of the two en^ sum will be the surface. Solidity = the area of the base x the perpendicular height Pyramid or Cone. Surface. — Multiply half the perimeter by the slant height, to w add the area of the base, the sum is the surface. Solidity = the area of the base x one-third the perpendic height. CBihlV.] IfBNSUaATION OF SOLIDS. 131 Pnutumofa Pyramid or Cone, Surface, — Multiply half the sum of the perimeters of the two ends hy the perpendicular height. SdidHy. — ^Add a diameter or side of the greater end to one of the leas, ind from the square of the sum subtract the product of the said two ditroeters or sides ; then multiply the remainder by one-third of the height, and this product by '785398 for circles, or by the proper Doitiplier for polygons ; the last product will be the capacity. That is, let D equal the greater diameter, d the less, h the perpen- ^Iw beight, and p = "785398 for cones, or for any pyramid the proper multiplier from column (a) in the table at page 118, then Solidity = jjoA(D* + Drf + rf«). SPHBBB. S^irface = the diameter x the circumference, = the square of the diimeter x 3*14159, = the square of the circumference X •3183. Sdiditp = the cube of the diameter x '5236, = the cube of the cbmifercnce x 01688. 8fkencd Segment. — Let d equal the diameter of the sphere, r the ndius of the base of the segment, and h its height ; then the otnednafaee = 3*14159 <f A. Solidiiy = 05236 k^ (3 </ — 2 A) ; or = 0-5236 A (3 r' -j- h"). The surface and solidity of a spherical zone may be obtained by ^ing the difference between the two segments. CONOIDS. A conoid is the solid generated by the revolution of a conic section Jj^nt one of its axes, and is called a epheroidy paraboloid^ or hyper- ^*fc*rf, according to the section from which it is produced. . Spheroids, — When the ellipse revolves about its transverse axis, it *• called an oblong or prolate spheroid; when about its conjugate axis, ^'^ (Hate spheroid; and when about any -other of its diameters, a ^^ivemd spheroi^ij in which latter case its figure is somewhat re- ^^ttbling a heart. To obtain the solidity of a spheroid, multiply the ?|^I^ttre of the revolving axis by the fixed axis, and the product '^ 0-5236. Pmdfoloid. — Let y be the radius of the circular base, and x the ^titode of the solid ; then Surface = ^i^^ {(/ + 4:r»)? - y^}. Solidity = V5708y^x. y^HptrboUnd, — Let t equal the transverse axis, r the radius of the ^*^ « the altitude, and p the parameter ; then the Solidity = J /^^ • . ^ K 2 132 MENSURATION OF SOLIDS. [PARr THE REGULAR OR PLATONIC SOLIDS. The regular or Platonic bodies are five in number, and have all been described at page 111. 1. To find either the surface or the solid content of any of M regular bodies. — Multiply the proper tabular area or surface (talf:^ from column (a) in the following table) by the square of the lin ^ edge of the solid, for the superficies. And Multiply the tabular solidity in column {h) of the table by the cvsl of the linear edge for the solid content. Surfaces and Solidities of Regular Bodies^ the side being unity or 1 No. of sides. Name. Surface. Solidity. (*) 4 6 8 12 20 Tetraedron Hexaedron Octaedron Dodecaedron Icosaedron 1-7320508 60000000 3-4641016 20-6457288 8-6602540 01178513 1-0000000 0-4714045 7-6631189 2-1816950 2. The diameter of a sphere being given^ to find the side of 4^ of the Platonic bodies^ that may be either inscribed in the sphere '9 circumscribed about the sphere^ or that is equal to the sphere, — ^^ tiply the given diameter of the sphere by the proper or correspond* number, in the following table, answering to the thing sought, ^^ the product will be the side of the Platonic body required. ^ The diam. of a sphere being 1 ; the side of a Tetraedron Hexaedron Octaedron Dodecaedron Icosaedron That may be inscribed in the sphere, is That may be cir- cumscribed about the sphere, is 0-8164966 0-5773503 0-7071068 0-3568221 0-5257309 2-4494897 1-0000000 1-2247447 0-4490279 0-6615845 That it equal to the sphere. 1-64394A0 0-8059958 10356300 0-4088190 0-6214433 3. The side of any of the five Platonic bodies being given^ to find the diameter of a sphere^ that may either be inscribed in thai body^ or circumscribed about it^ or that is equal to it, — As the respective number in the table above, under the title inscribed^ eircumscribedy or equcdy is to 1, so is the side of the given Platonic body to the diameter of its inscribed, circumscribed, or equal sphere. 4. The side of any one of the fioe Platonic bodies being given^ to find the side of the other four bodies, that may be equal in solidi^ to that of the gioen body. — As the number under the title equal in the :aiP. IV.] MENSURATION OP SOLIDS. 133 mMl colomn of the table above, against the given Platonic body, is to be number under tbe same title, against the body whose side is ^agfat, 80 is tbe side of the given Platonic body to the side of the M)dj sought. Besides these tbere are thirteen demiregular bodies, called Solids ^Archimedes, Tbey are described in the Supplement to Lidonne's f^M$ de tons les Diviseurs des Nombres^ &c., Paris, 1808 ; twelve »f them were described by Abraham Sharp, in his Treatise on E^olycdra. re nND THE CONTENTS OF SURFACES AND SOLIDS NOT REDUCIBLE TO ▲BY KNOWN FIGURE, BY THE EQUIDISTANT ORDINATE METHOD. The general rule is included in this proposition, viz. : — If any ri^btline be divided into any even number of equal parts, 1, 2, 3, 4, &C., (fig. 101,) and at the points of division be erected perpendicular oxdiDates 1 A, 2 B, 3 C, &c., terminated by any curve A C G : then, iC a be put for the sum of the first and last ordinates, 1 A, 7 O, e for die sum of the even ordinates, 2 B, 4 D, 6 F, &c., viz., the second, fourth, sixth, &c., and o for the sum of all the rest, 3 C, 5 E, &c., "VIZ., the third, fifth, &c., or the odd ordinates, excepting the first and la«t: then, the common distance 12, 2 3, &c., of the ordinates l>cing multiplied into the sum arising from the addition of «, four ^TDes e, and twice o, OTte third of the product will be the area 1 A G 7, ▼wy nearly. _ / # + 4^ -h 2o That 18, — - — --^ . D = area, D being = A C = CE, &c. 3 The same theorem will equally serve for the contents of all solids, by oaog the sections perpendicular to the axis instead of the ordi- iistet. The proposition is quite accurate, for all parabolic and right ^ areas, as well as for all solids generated by the revolutions of ^ic sections or right lines about axes, and for pyramids and their ^Wims. For other areas and solidities it is an excellent ap- proximation. The greater the number of ordinates, or of sections, that are taken, ^ more accurately will the area or the capacity be determined. Bot b a great majority of cases^t^e equidistant ordinates, or sections, ^ lead to a very accurate result. 134 PLANE TRIOONOMBTRY. [PAKT I. CHAP. V. TBiaONOMETRY. Sect. I. Definitions and lVi</onotnetriciU Formulae. 1. Plane Trigonometry is that branch of mathematics by which we learn how to determine or compute the unknown parts of a plane, or rectilinear triangle, from those which are known, when that 18 possible. Every triangle consists of six parts, viz , three sides, and three angles opposite those sides. And any three of these being given (excepting only when the three angles are given) the others may always be determined from them. The determination of the mutual relation of the sincs^ tangents, secants^ &c., of the sums, differences, multiples, &c., of arcs or angles; or the investigation of the connected formulae, is also usually classed under plane trigonometry. 2. Let ACB (fig. 102) be a rectilinear angle: if about C as a centre, with any radius C A, a circle be described, intersecting C A, CB, in A, B, the arc AB is called the measure of the angle ACB. 3. The circumference of a circle is supposed to be divided or to be divisible into 360 equal parts, called degrees; each degree into GO equal parts, called minutes; each of these into 60 equal parts, called seconds; and so on to the minutest possible subdivisions. Of these, the first is indicated by a small circle, the second by a single accent, the third by a double accent, &c. Thus, 47° 18' 34" 45''', denotes 47 degrees, 18 minutes, 34 seconds, and 45 thirds. The number of degrees, minutes, seconds, &c., contained in the arc AB of the circle described from the angular point C, and which is contained between the two legs AC, A B, is called the measure of the angle ACB, which is then said to be an angle of so many degrees, minutes, seconds, &c. Thus, since a quadrant, or quarter of a circle, contains 90 degrees, and a quadrantal arc is the measure of a right angle, a right angle is said to be one of 90 degrees. 4. The complement of an arc is its difference from a quadrant, as B E ; and the complement of an angle is its difference from a right angle, as £ C B. 5. The supplement of an arc is its difference from a semicircle, as A' £ B ; and the supplement of an angle is its difference from two right angles, as A' C B. CHIP, v.] PJLANB TBIOONOIIBTKY. 135 6. The tine of an arc is a perpendicular let fall from one extre- nitj opon a diameter passing through the other, as B D. 7. the versed eine or versine of an arc is that part of the dia- eeter which is intercepted between the foot of the sine and the arc, uDA. 8. The tangent of an arc is a right line which touches it in one ex- tRmity, and is limited by a right line drawn from the centre of the drde through the other extremity, as A T. 9. The tecant of an arc is the radial line which thus limits the tugent, as C T. 10. These are also, by way of accommodation, said to be the sine, tasgent, &c., of the angle measured by the aforesaid arc, to its deter- mintte radios. 11. The cosine of an arc or angle, is the sine of the complement of that arc or angle, as G B : the cotangent of an arc or angle is the tiDgent of the complement of that arc or angle, as E M. And the OHxntd sine E O, and cosecant CM, are similarly the versed sine and secant of the complement. 12. The suversine of an arc is the versed sine of its supplement, MAD. ^flte.— The following contractions are employed to express the foregoing terms, viz. : — For the radius of the arc A B we write rad A B, sine ditto sin AB, tangent ditto tan AB, secant ditto «^c A B, versine ditto versin AB, cosine ditto om AB, cotangent ditto cot AB, cosecant ditto cosec AB, coversine ditto covers AB. 13' The /oiiowing Corollaries may he drawn from the above Definitions: — (A.) When the arc is evanescent, the sine, tangent, and versed ■Be^ are evanescent also, and the secant becomes equal to the radius, that being its minimum limit. As the arc increases from this state, the snes, tangents, secants, and versed sines increase ; thus they con- tnme till the arc becomes equal to a quadrant A £, and then the sine K m its maximum state, being equal to the radius, and is then called ^ sine total; the versed sine is also then equal to the radius ; and ^ tecaot and taneent becoming incapable of mutually limiting each <Hher, are regarded as infinite. Id employing these lines for the purposes of calculation, they are ■D eonsidercKl as htmng positive values for any arc not exceeding 90®; hit in the second quadrant, the cosine falling on the opposite side of the diameter, and being measured in an opposite direction, is con- 'i^ered negative^ but the sine remains positive; in the third quadrant 136 PLANE TRIGONOMETRY. [part I. the cosine is still negative^ and the sine, having now changed its direc- tion, is negative also ; in the fourth quadrant, the cosine having again returned to that side of the diameter on which it was in the first quadrant, again hecomes positive^ hut the sine remains neg€Uive. The signs of the others are determined by the ordinary rules of algebra, from the formulae at page 1 37. The following table exhibits both the value of trigonometrical lines at the commencement of each qua- drant, and also the signs with which they are affected in passini through the same. Value atO«. Sign in Ist Quad. Value atOO*. Sign in 2nd Quad. + 1 Value' atl80*. _ O Sign in 3rd Quad. Value at 270*. Sign in 4th Quad. Value at aeo*. Sin ... O -f R R o Tan... O -f 00 — o + 00 — o Sec ... R + 00 - R — 00 + R Versin O + R ■f 2R + R + o Cos ... R + O - R — O + R Cot ... 00 + O — « -i- O — 00 Cosec 00 + R -f 00 - 1 R — 00 R signifies equal to rad ; oo — infinite ; — evanescent. (B.) An arc and its supplement have the same sine, tangent, anc secant. (C.) Of any arc less than a quadrant, the arc is less than its cor- responding tangent; and of any arc whatever, the chord is less thai the arc, and the sine less than the chord ; but the smaller the arc, th< nearer they all approach to equality. (D.) The sine BD of an arc A B, is half the chord BF of th< double arc B A F. (£.) The versed sine of an arc, together ^4th its cosine, are equa to the radius. Thus, AD -fBG=AD + DC = AC. (F.) The radius, tangent, and secant, constitute a right-angle< triangle CAT; and the cosine, sine, and radius, constitute anothei right-angled triangle C D B, similar to the former. So, again, the co- tangent, radius, and cosecant, constitute a third right-angled triangle MEG, similar to both the preceding. Hence, when the sine anc radius are known, the cosine is determined by the property of the right-angled triangle. The same may be said of the determination ol the secant, from the tangent and radius, &c. &c. &c. (G.) Further, since the triangles CAT and MEC are similar A T : CA : : B D : D C : : C E : E M, or tan ; rad : : sin : cos : : rad : cot • Also C T : C A : : C B : C D : : C M : £ M, or sec : rad : : rad . : cosec : cot. And CM:CE::CB:DB or cosec : rad : : rad : sin. cos CHAP, v.] PLANS TBIGONOMBTBY. 187 (H.) Also, by Geom. Sect. V. Prop. XXII., as AD : DB : : DB : HD, or sin* = H D . versin ; or since H D = pad -|- cos, tben sin^ tersin = rad + cos (I.) From Arts. (F.) (G.) and (H.) we deduce the following fonnolsB :— 1. sin = >/Tad* — cos* = . cosec 2. tan := >/sec* — rad* = 3. sec = v^rad^* + tan* = cos rad . sin rad^ cos cot rad» 4. versin = rad + cos * 5. cos = >/rad'^ — sin* = sec 6. cot := V cosec' — rad""* = rad« c rad . cos rad^ sin tan 7. cosec =: Vrad* + cot* = -;— . sin cos* 8. covers = • rad + sin* CK.) If unity be regarded as tbe radius of the circle, the above fonnoijg become : — 1. sin = >/l — cos* = . cosec / — 5 sin 1 2. tan = V sec* — 1 := — = — - . cos cot 3. sec = >/l + tan*= — . cos sin* 4. versm = 1 + cos 5. cos = >/ 1 — sin* := — . sec cos cot = V cosec? — 1 = -r-^ : — sm tan 138 PL^NB TBIOONOMBTBY. [PABT I. 7» cosec = >/l + cot* = — r- . sm COS* 8. covers = :-. 1 + sm 14. The following are some of tbe most useful formulsB relating to trigonometrical lines, in all of which radius is regarded as unity. (a.) Eapressions/or the sine and cosine : — Let a equal tbe length of the arc, then "'"* = ''-r:T:3 + 1.2.3.4.5 - 1.2.3.1.5 6.7 +^-^'^ oosa=l-^+P^^L-^- ^^ /^^^ +&c (2). (6.) Expressiofis for the sum and difference of two arcs: — Let a and h be the two given arcs, then sin (a 4- 5) = sin a cos 6 + cos a sin b (3) sin (a — 5) = sinacos b — cosasin b (4) cos (a 4- &) = cos a cos 5 — sin asin b (5) cos (a — ft) = cos a cos ft + sin a sin 6 (6) , ,. tan a + tan ft . tan(a -f ft) = , --^ — t (7) ^ ' 1 ■— tan a tan ft ^ tana — tan ft tan(a-ft = — — — j- (8) "^ I -H tan a tan ft ^ ' cot a cot ft — i cot(a + ft) = TTT^ — 7— (9) ^ ' cot ft + cot a ^ ^ „ cot a cot ft + 1 , ^ cot(a - ft) = -r -— (10). "^ ' cot ft — cot a ^ ' (c.) Expressions for the sine and cosine of mtdtiple arcs: — Let n be any integral number, then sin «a = cosa{(2sina)-» -^^(2sina)-^ + (!!Z?K^i:D(2sina)-^ - """!°.i.y"^ '-"A^<^c) ,„) cosnasz: J{(2sina)" — n(2sina)— * + ^ ? "l^V ^sina)-^ n(n — 4)(« — 5V . , . , 1.2 3 ^^(28ina)-« + &c.} (12 CHAP, v.] PLANl TBIOONOMBTBY. 139 (c^) Table o/mMkipk arcs : — Let « be the ain, t the tan, and e the cot of the arc a, then 1. sin 2"V 2 > 2. 8ina=:«, 3. 8m2a = 2« v^l — «*, 4. m3a = Bs—4f\ 5. 8in4a = (4* — 8r^ >/l — r*^ 6. 8m5a = 16#' — 20*' + 5«, 13.tan- = -'_, 2 n.>/i^.^' 7. cos 2' '5. tan 2a = ^^ Unda = ^^- tan4a = ^®- tan 5a = 2/ 1-3/** 4^ — 4/» 1— 6/» + /*' 1 — 10/« + 5^' 8. cos a ^ Vl — a", 9. cos 2a = 1 —2a', 10. cos 3a = (1 — 4«^) Vl— r', 11. co84a = l — 8*«^-8«^ 12. cos5a-(l-12**+16«*)Vr^. « 1 19. cot- = , , 20. cot a =e, 21. cot2a=: 22. cot 3a = 23. cot 4a = 24. cot 5a = c^-1 c* — 3<? 3c«- 1' c*^6c'+ 1 4(r'-4c ' c^— 10c3 -i- 5c 5c*— 10c« + 1 " 5s. C ^') Table of Powers of the sine and cosine. un as una, Sain'aa 1 .-.oo«2a, liin^a"" Ssina — sin So, Sun* a« 3 — 4 COS 2a 4- COS 4 a, )6uii*as sin 6 a— 6 sin 3a4- lOsin a, 6. cos aBCOsa, 7. 2008*0-* 0062a + I9 8. 4oos'a»co8 3a + 3oosa, 9. 8G08*a»oos4a + 4oos2a + 3, 10. 16co8*aB0066a-f5oos3a-i-10co8a. ». ^/) Expressions for the arc in terms of the sine or tangent : — ^^t « be the sin, and t the tan of the arc a, then (13) tKa = t — - + -—6ie (14). SscT. II. Trigonometrical Tables. From the forcing, and other properties, and theorems, mathe- ba¥e computed the lengths of the sines, tangents, secants, 140 PLANE TRIOONOMBTBY. [p^ .t 1 and versed sines (assuming unity for the radius) corresponding to w from 1 second of a degree, through all the gradations of magni^^ad up to a quadrant, or 90^ ; and the results of the computation^^ t arranged for use in tahles called Trigonometrical Tablet. As, B=ioi ever, these quantities have to he carried to several places of decSL mi in order to ohtain sufficiently accurate results, their use in cal ^cali tions is attended with much lahour, and therefore it is usual to enr~^plc their logarithms instead ; hut in this case the assumed radius is t-^ce as 10,000000000 instead of unity, since with the latter most o^^ th quantities would he fractional^ and therefore have negative ch wuwt, teristics, the use of which would be inconvenient, and is supers^e by taking the radius as above. Table IV. in the Appendix is such a table of the logarithiKBS < the sines, cosines, tangents, cotangents, secants, and cosecants^ f< every minute from 1 minute to 90 degrees, calculated to a radius < 10,000000000 as above. It will be observed that the headm^s * the columns run along the tops of the pages as far as the 45th de^i"^ after which they return along the bottoms of the pages in contra order, as below : — sin D. cosec tan D. cot sec D. co« cos D. sec cot D. tan cosec 1 D. &C^ The reason of this will be apparent, if we only consider that the cc^ cot, or cosec of an arc, is the sin, tan, or sec of the complement ^ that arc. The intermediate columns, headed D, contain the diffe^ ences of the consecutive logarithms in the contiguous columns o '^ either side ; it will be seen that the same difference is common to thti^ sin and cosec, the tan and cot, and the sec and cos; since fron^ rad* rad*^ Art. (I.) page 137, sin = — —^ or rad^ = sin . cosec; tan = — -^ or rad ^ = tan . cot ; sec = cosec rad 2 cot cos or rad '^ = sec . cos ; and conse- quently, log sin + log cosec s= log tan 4" log cot = log sec + log cos = 2 log rad = 20* ; therefore as the sin, tan, or sec increases, so must the corresponding cosec, cot, or cos, diminish, and their dif- ferences must be equal. USB OP TABLE IV. To find the logarithmic sine^ tangent^ Spe, of a given are. If the arc contains only degrees and minutes, its sin, tan, &c., will be found simply by inspection, by looking along the top or bottom of the tables for the degrees, and then in the first or last vertical column for the minutes, according as the number of degrees is less or greater than 45 ; and on the same line, in the column having for its title .v.] PLANE TRIOONOMBTRT. 141 ler At the top or bottom, according as the degrees were found) the z of the trigonometrical quantity required, its log will be found. f the arc contains seconds as well, the logarithm must be found as ife for the degrees and minutes ; then take the number in the otigooas column headed D on the same line, multiply it by the nnberof seconds, and divide by 100 (which is done by cutting off le two list figures) ; the quotient must then be added to or subtracted RND the log already taken out, according as the same would be in- aeued or decreased by an increase in the arc. Examples, 1. Find the log sin of 37° 47'. As the arc is less than 45°, by looking along the top of the table for the degrees, and in the^r«/ column for the minutes, we find in the colomn baying at its top the word sin, the figures 9*787232, which is the log sin of the arc required. 2. Find the log tan of 75° 34'. Here, as the arc is greater than 45°, looking at the battom of the tihles for the degrees, and in the last column for the minutes, we find in tbe column having tan at the bottom, 10*589431, the tan of rS'' 34'. 3. Find the log sin of 31° 45' 5". Tie log sin of 31° 45' is . . . . 9-721 162 I The No. in col. D is 3402 x 6" -f- 100 = + 1 70 | 10 .-. The log sin of 31° 45' 6" = . 9-721332 4. Find the log cos of 25° 1' 47". Tbelogcos of 25° 1' is .... 995721 7 The No. in col. D is 983 x 47" h- 100= ^ 462 01 •. The log cos of 25° 1' 47" = . . 9956755 Bb Tojind the are corresponding to any given log sin, tan, &c. : — ^km Table IV. for the given log sin, &c., or the next less log ^''^'^ and on the same line will be found the minutes, and at the ^P or bottom of the page the degrees of the arc required ; if the log ^ foond is less than the given log, subtract the former from the p^i add two cyphers to the right of the remainder, and divide it 7 the nnmber found in the contiguous column headed D ; the quo- ^Qt will he the nnmber of seconds to be added to the degrees and ^tes in the arc already obtained. Examples. J* Fbd the arc wboae log tan is 10*577537. Here the arc is found by inspection to be 75° 11'. 142 PLANB TBIQONOMBTBY. [PABT I. 2. Find the arc corresponding to the log sin 9*395401. Given log == 9-396401 Next less log = 9-396166 = log tan 14*> 23' 23500 -f- 492 = 48 seconds ; .-. 9-396401 is the log tan of 14° 23' 48", In tbe sines and tangents of arcs less than about 2°, the differences between any two successive values are so great (as will be seen by an inspection of column D in tbe table), that the method above given for finding the intermediate values for seconds will not be sufficiently correct ; and the same remark applies to the cosines and tangents of arcs greater than about 88°. It will also be observed, that in the cosines and secants of arcs less than 2°, and in the sines and cosecants of arcs greater than 88°, the differences are too small to enable us to calculate accurately the value of any arc from them. The first of these difficulties may be removed by the rules given below for determining the values of the sines and tangents of small arcs, and the tangents of large arcs, and conversely the arcs from the sines and tangents. The second difficulty, however, could only be got over by extending the tables to more decimal places, but as this would also require all other Quantities employed in the same calcula- tions to be taken to an equal number of decimals, much additional trouble would be occasioned ; and it is therefore better for determin- ing the value of an arc when near 90°, to employ some other function than its sine, as, for instance, its cosine. In order to render this clearer to those who are not familiar with the use of logarithms, we subjoin an example of such a substitution of the cosine for the sine. For instance, let it be desired from the formula, Pj : Pg • : sin /9 : sin ^ to determine the value of the angle ^, when P j = 600-1 ; P^ = 669; and the angle = 63° 45'. First by multiplving the means and extremes, and dividing both sides by P, , we obtain . . Pnsinfi sm ^ = p (a). Then from Art. (L), page 137, sin ^ = >/ 1 — (cos iy Pj sin $ = N/l-.(cosd)«; •^1 squaring both sides / p, sin e y {^-^ — J = 1 - (cos ay, CKJIP.?.] PLANB TBIOONOMBTRT. 143 trmssponng, and eztnctiiig the square root cos ,=V-(^)- («)• We haYe, therefore, two equations (a) and (^), from either of 'vrlieh we can ohtain the yalae of the angle ^, hy substituting the iralQes of the known quantities ; but in doing so we shall find that the second equation will give the yalue of J much more exactly than tliefint. Thua^by tubstitnting the known quantities in equation (a), we have 669 X sin 63° 45' . . -— — = sm i. 6001 Whence by logarithms. Log sin ea** 45' =r 9-952731 Log 669 = 2-825426 12-778157 Log 600-1 = 2-778224 Log sin i = 9-999933 .-. i = 88<> 59' 25". Phiceedmg in a similar manner with equation (5), we have /669 X sin 63'' 45'V ^-C — 6001 — ;=<^-^- Wbeoce by logarithms, taking the radius as unity, for the reasons wpliined at page 146, Log sin 63° 45' = 1952731 Log 669 = 2-825426 Log 600-1 2-778167 = 2-778224 T-999933 2 Loff -99969 = r999866 Then I ^ -99969 = -00031, the log of which = 4-491362, and 4-491362 -r 2 = 2-245681 = cos i; or restoring the radius of the tables, cos ^ = 8-245681 = sin of the complement of i ; 144 PLINB TRIOONOMBTBY. [PABT I. therefore, bj the rule given below for finding a small arc from its sin, we have 8-246681 6-314425 22 3-560128 = 3631-86 seconds = r 0' 31" -86 ; ^^=^ . ^ _ ggo 59/ 2g// .14^ whence we see that the former value of J obtained from equation (a) is upwards of three seconds too small. To find nccurately the log sin of an are less than 2°. Reduce the arc to seconds, and find the log of that number from Table III., to which add 4-685575 (the log sin of 1'), and subtract one-third of the decimal portion of the log sec of the arc taken from Table IV. ; the remainder is the log sin of the arc required. Examples. Find the log sines of 13' and of 1° 3'. Log (13' X eW) = 780" - 2-892096 4-685575 7-577670 Log8ecofl3'»-000003-r3» -000001 Log sin of 13' « 7-577669 Log (ey X 60) « 3780" « 3-577492 4-685575 8-263067 Logsecof r3'»-000073^3» -000024 Log sin of 1° y ^ 8-263043 To find accurately the log tan of an arc less than 2®. To the log of the number of seconds in the arc add 4-685575, and two-thirds of the decimal portion of its log sec; the sum is the log tan of the arc required. Examples, Find the log tans of 24' and of 1° l.V. Log (24' X 60} -: 1440" <= 3158363 4-085575 Log8ecof24'«*000011x}" '000007 Log tan of 24' « 7843945 Log (75^ X 60) - 4500" » 3-653213 4-685575 Log sec of ri5'»=-000l03x |» -000068 Log Un of r 15^ « 8-338856 To find accurately the log tan of an arc greater than about 88®. Add to the log of the number of seconds that the arc is less than 90% two-thirds of the decimal portion of the log cosec, and subtract the sum from 15*314425; the remainder will be the log tan required. Examples. 1. Find the log tan of 89° 5' 13". 90'' 0' 0" 89 5 13 0^ 64' 47" = 3287 seconds CBAP.V.] PLANS TRIOONOMBTBY. 145 Constant log .... = 15-314426 Log of 3287 . . . ; = 3-6 16800 Log cosec of 89« 5' 13'' = -000057 x | = -000038 3-516838 Log tan of 89'' 5' 1 3'' = 11797587 a. Find the log tan of SS'' 61' 10". 90° 0' 0" 88 51 10 r 8' 50" = 4130 seconds. CoMtMtlog =15-314425 Ii0gof4130 =3-615950 Xogcoscc of 88** 51' 10" = -000087 x | = -000058 3-616008 Log tan of 88° 61' 10" = 11-69 8417 To find accurately an arc of not more than il°/rom its log sine, Totbegiyen log sin, add 5*314425, and one-third of the decimal poition of the secant of the nearest arc to that whose log sin is given, ^« som, rejecting 10 from the characteristic, will be the logarithm ^^ tke number of seconds in the arc. Example. Heqnired the arc whose log sin is 8314719, GiTen log sin . . . = 8-314719 Constant log .... = 5-314425 Ug sec of nearest arc = -000093 -f- 3 = -000031 Arc required 1° 10' 58" = 4258" = 3-629175 To find accurately an arc of not more than ^^from Us log tan, To the giTcn log tan add 5*314425, and from the sum subtract ^''^[tJWs of the decimal portion of the log sec of the arc whose log ^i« nearest to that given, and the remainder, rejecting 10 from the ^''ctemtic, will be the log of the number of seconds in the arc. Example, Required the arc whose log tan is 8*231461. Given log tan . . . . = 8*231461 Constant log . . . . = 6*314425 13-545886 Log sec of nearest arc = -000063 x | = -0000 42 Required arc = 58' 34" = 3514" = 13-546844 L 146 GENEBIL PROPOSITIONS. [PABT I. To find (accurately an arc greater than S^"^ from its log tan. Add to the given log tan two-thirds of the decimal portion of the log cosec of the nearest arc to that whose log tan is given, and sub- tract the sum from 15*314425, the remainder is the log of the num- ber of seconds that the arc is less than 90^. Example, Required the arc whose log tan is 11*695900. Constant log ... . = 15-814425 Given log tan .... =11-695900 Log cosec of nearest arc 000088 X | = -000059 11-695959 Required arc = 1° 9' 14'' = 4154" = 3-618466 ON THE ROOTS AND POWERS OF TRIGONOMETRICAL QUANTITIES. In extracting the root, or raising the power of any trigonometrical quantity by means of its logarithm, it will always be found most con- venient to reduce the assumed radius to unity, by subtracting 10 from the characteristic of the logarithm, observing where the characteristic thus becomes negative the rules given at pages 47 and 48 An ex- ample of this alteration of the charactenstic is given at page 143. TRIGONOMETRICAL QUANTITIES OP ANGLES GREATER THAN 90*^. Although Table IV. only purports to give the sines, tangents, &c., of angles less than, or equal to OO**, any of these functions of angles greater than 90° may readily be obtained, since any function of an angle greater than 90° is equal to the same function of the supple- ment of that angle ; due regard being had to changing the sign, if requisite, according to the table given at page 136. Thus the log sin of 141° 15' is 9-79052 1, the same as the sin of its supplement 38° 45'; the log sec of 95° 43' is — J1001701, or the sec of 84° 17' with its sign changed ; the log cosec of the same is I0-00S165, or the cosec of 84° 1 7', the sign remaining the same ; and the log tan of 173° 4' is — 9 084947, or the tan of 6° 56', with its sign altered. Sect. III. General Propositions, I. The chord of any arc is a mean proportional between the versed sine of that arc and the diameter of the circle. II. As radius is to the cosine of any arc, so is twice the sine of that arc to the sine of double the arc. III. The secant of any arc is equal to the sum of its tangent, and the tangent of half its complement. IV. The sum of the tangent and secant of any arc, is equal to the tangent of an arc exceeding that by half its complement. Or, the / CfliP. v.] OBNBBAL PROPOSITIONS. 147 9am of the tangent and secant of an arc is eqoal to the tangent of 45'/>/i«half the arc, V. The chord of 60^ is eqaal to the radius of the circle; the ^vereed sine and cosine of 60° are each equal to half the radius, and the secant of 60° is equal to double the radius. VI. The tangent of 45° is equal to the radius. VII. The square of the sine of half any arc or angle is equal to a rectangle under half the radius and the versed sine of the whole arc; and the square of its cosine is equal to a rectangle under half tiie radios and the versed sine of the supplement of the whole arc or aogle. nil. The rectangle under the radius and the sine of the sum or tile difference of two arcs is equal to the sum or the difference of the i^Bctangles under their alternate sines and cosines. IX. The rectangle under the radius and the cosine of the sum or ^e difference of two arcs, is equal to the difference or the sum of the rectangles under their respective cosines and sines. X. As the difference or sum of the square of the radius and the '^^ctaogie under the tangents of two arcs, is to the square of the '^xiios; so is the sum or difference of their tangents, to the tangent of t^ sum or difference of the arcs. •XI. As the sum of the sines of two unequal arcs, is to their dif« ^f^nce; so is the tangent of half the sum of those two arcs to the ^Sigent of half their difference. ^11. Of any three equidiffereut arcs, it will be as radius is to the ^**«ne of their common difference, so is the sine of the mean arc, to "^If the sum of the sines of the extremes ; and, as radius is to the 1^X1 c of the common difference, so is the cosine of the mean arc to "^If the difference of the sines of the two extremes. 4^A.) If the sine of the mean of three equidiffercnt arcs (radius ■^^ing unity) be multiplied into twice the cosine of the common dif- ^^*^ce, and the sine of either extreme be deducted from the pro- ^'ict, the remainder will be the sine of the other extreme. (B.) The sine of any arc above 60°, is equal to the sine of an- other arc as much below 60°, together with the sine of its excess •^Tc 60°. ^. From this latter proposition, the sines below 60° being known, **^<»e of arcs above 60** are determinable by addition only. ^ni. In any right-angled triangle, the hypothenuse is to one of |he legs, ag the radius is to the sine of the angle opposite to that ^^; and one of the legs is to the other as the radius is to the tan- S^t of the angle opposite to the latter. . ^IV. In any plane triangle, as one of the sides is to another, so ^ the sbe of the angle opposite to the former to the sine of the ^^^ opposite to the latter. ^^* In any plane triangle it will be, as the sum of the sides Jjj^t the vertical angle is to their difference, so is the tangent of '^ the sam of the angles at the base, to the tangent of half their «wrence. l2 148 SOLUTION OP THK CASES OP PLANE TBIANOLBS. [PABT I. XVI. In any plane triangle it will be, as the cosine of the dif- ference of the angles at the base, is to the cosine of half their sum, so is the sum of the sides about the vertical angles to the third side. Also, as the sme of half the difference of the angles at the base, is to the sine of half their sum, so is the difference of the sides about the vertical angle to the third side, or base *. XVII. In any plane triangle it will be, as the base is to the sum of the two other sides, so is the difference of those sides to the dif- ference of the segments of the base made by a perpendicular let fall from the vertical angle. XVIII. In any plane triangle it will be, as twice the rectangle under any two sides, is to the difference of the sum of the squares of those two sides and the square of the base, so is the radius to the cosine of the angle contained by the two sides. Cor, When unity is assumed as radius, then if A C, A B, B C, are the sides of a triangle and C the angle opposite the side A B, this AC^ -h BC^ — AB- , . ., Prop, gives cos C = — - : and similar expressions <« C B . C A for the other angles. XIX. As the sum of the tangents of any two unequal angles is to their difference, so is the sine of the sum of those angles to the sine of their difference. XX. As the sine of the difference of any two unequal angles is to the difference of their sines, so is the sum of those sines to the sine of the sum of the angles. These and other propositions are the foundation of various for- mnlsB, for which the reader who wishes to pursue the inquiry may consult the best treatises on Trigonometry. Sect. IV. Solution of the C<we« of Plane Triangles, Although the three sides and three angles of a plane triangle, when combined three and three, constitute twenty varieties, yet they furnish only three distinct cases in which separate rules are required. CASE I. When a side and an angle are two of the given parts. The solution may be effected by Prop. XIV. of the preceding sec- tion, wherein it is affirmed that the sides of plane triangles are re- spectively proportional to the sines of their opposite angles. In practice, if a side be required, begin the proportion with a sine, and say, As the sine of the given angle. Is to its opposite side ; So is the sine of either of the other angles, To its opposite side. * Thene propositions were first given by Thacker in his Mathematical Miaeel- lanjfy published in 1743 ; their practical utility haM been recently shown by Pro- feMvr Wallac€t in the Edinburgh Philosophical TraruacHons. AV.Y.] SOLUTION OF TUB CASKS OF PLANE TRIANGLES. 149 If an M^ he required, begin the proportion with a side, and As one of the given sides, Is to the sine of its opposite angle ; So is the other given side, To the sine of its opposite angle. The thud angle becomes known by taking the sum of the two former from 180°. NUe. — It is usually best to work the proportions in trigonometry bj means of the logarithms, taking the logarithm of the Jirst term from the sum of the logarithms of the 9econd and thirds to obtain the logarithm of the fourth term. Or, adding the arithmetical comple- ifiAi/of the logarithm of the first term to the logarithms of the other two, to obtain that of the fourth. CASE II. When two sides and the included angle are given. The solution may be effected by means of Props. XV. and XVI. of die preceding section. Thus: take Uie given angle from 180% the remainder will be the sum of the other two angles. TlienMy, As the sum of the given sides, Is to tlieir difference ; So is the tangent of half the sum of the remaining angles. To the tangent of half their difference. Then, secondly say, As the cosine of half the said difference. Is to the cosine of half the sum of the angles ; So is the sum of the ffiven sides. To the third, or required side. As the sine of half the diff*. of the angles, Is to the sine of half their sum ; So is the difference of the given sides, To the third side. ^•tmjU.^hi the triansle ABC (Fig. 9.) are given A C = 450, ^C 3 540, and the included angle C c= 80"* ; to find the third side, ^ the two remaining angles. Then, Log(BC -AC as 90)= 1-954243 Log tan (i A + B = 50<>) = 10076187 12030430 Log(BC-f'AC =: 990) = 2-995685 Log tan (4 A - B =c 6^ 11") = 9034795 150 SOLUTION OF THE CASES OF PLINE TRIANGLES. [PlBT I. Then, Log cos (| A -i- B = 50°) = 9-808068 Log (BC -h A C = 990) = 2-995635 12-803703 Log cos (i A — B = 6° 1 V) = 9-997466 Log(AB ... =640-08)= 2-806237 Also, ^(A -i-B) + HA-B) = 56° ir=A; and J (A -f B) _ J (A - B) = 43° 49' = B. Here, much time will be saved in the work by talcing log cos J (A -h B) from the tables, at the same time with log tan ^ (A -h B) ; and log cos |(A — B) as soon as log tan ^ (A — B) is found. Ob- serve, also, that the log of B C -I- A C is the same in the second operation as in the first. Thus the tables need only be opened in Jive places for both operations. A nother solution to Case II, Supposing C to be the given angle, and C A, C B, the given sides ; then the third side may be found by this theorem, viz. ; — A B = ^/(A C^ -f B C'' — 2 A C . C B . cos C). Thus, taking A C = 450, B C = 540, C = 80% its cos -1736482 AB = ^(450^^+ 540'^- 2 x 450 x 540 x -1736482) = ^{90=^ (5- + 6'- 2 X 5 X 6 X -1736482)} = 90 ^50-58118 = 90 X 7-112 = 64008, as before. CASE III. When the three sides of a plane triangle are given^ to find the angles, 1st Method. — Assume the longest of the three sides as base, then say, conformably with Prop. XVI., last section, As the base. Is to the S!im of the two other sides ; So is the difference of those sides. To the difference of the segments of the base. Half the base added to the said difference gives the greater seg* ment, and made less by it gives the less ; and thus, by means of the perpendicular from the vertical angle, divides the original triangle into two, each of which falls under the first case. 2nd Method. — Find any one of the angles by means of Prop. XVIII. of the preceding section ; and the remaining angles either by a repe- tition of the same rule, or by the relation of the sides to the sines of their opposite angles, viz. : — A C^ + B C2 — A B* A B'^ + B C — A C* cos C = ; cos B = ir7-~——^ 2AC.BC 2AB.BC B A^ + A C^ — B C« and cos A = . 2AB . AC r^HAP. v.] SOLUTION OP THB CASES OF PLANA TRIANGLES. 151 RIGHT-ANGLED PLANE TRIANGLES. Right-angled triangles may (as well as others), be solved by means if the rule to the respecUye case under which any specified example alls; and it will then be found, since a right angle is always one of lie data, that the rule usually becomes simplified in its application. When two of the sides are given, the third may be found by neans of the property in Geom.^ Prop. XVI. Sect. III. Hypoth. = v^fbase^ -|- P^rp-^) Base = >/(hyp.* — perp.*) = >/(hyp. + perp.) . (hyp. — perp.) Pcrp. = ^(hyp.^ — base^) = >/(hyp. + base) . . (hyp. — hase). There is another method for right-angled triangles, known by the phnse making any Me radius ; which is this. ** To find a side. Call any one of the sides radius, and write upon it the word radius ; observe whether the other sides become sines, tansents, or secants, and WTite those words upon them accordingly. Call the word written upon each side the name of each side: then say, As the name of the given side, Is to the given side ; So is the name of the required side, To the required side." ^ To find an angle. Call either of the given sides radius, and vrite upon it the word radius; observe whether the other sides ^^^come sines, tangents, or secants, and write those words on them accordingly. Call the word written upon each side the name of that "We. Then say, As the side made radius. Is to radius ; So is the other given side. To the name of that side, vhich determines the opposite angle." When the numbers which measure the sides of the triangle are <i^ nnder 12, or resolvable into factors which are each less than 1^) the solution may be obtained, conformably with this rule, easier vithoQt logarithms than with them. For, Ut ABC (Fig. 103) be a right-angled triangle, in which A B, the hose, is assumed to be radius ; B C is the tangent of A, and A C te secant, to that radius ; or, dividing each of these by the base, we "ifl ha?e the tangent and secant of A, respectively, radius being ^tj. Tracing in like manner the consequences of assuming BC (tt fig. 104), and AC (as fig. 105), each for radius, we shall readily •l>tMn these expressions. 1. ^p^* = tan angle at base. (Fig. 103.) 2. -^ = sec angle at base. (Fig. 103.) base 152 HEIGHTS AND DISTANCES. [PART 3. = tan angle at vertex. (Fig. 105.) 4. -^-^ = sec angle at vertex. (Fig. 105.) 5. -r—^' = sin angle at base. (Fig. 104.) base _, 6. = sm angle at vertex. (Fig. 104.) Sect. V. On the application of Trigonometry to measuring HeighU and Distances. Trigonometry receives its principal practical application in the operations of surveying, and measuring heights and distances; as, however, the methods of its application (depending on the peculiar circumstances of each case) are exceedingly various, we cannot lay down any general rules, but must content ourselves with giving a selection of such examples as are most likely to occur ; and the prin- ciples developed in which, will be sufficient to guide any person in its further application to other cases. The instruments employed to measure angles are quadrants, sex- tants, theodolites^ &c., the use of either of which may be sooner learnt from an examination of the instruments themselves than from any description independently of them. For military men and for civil engineers, a good pocket sextant, and an accurate micrometer (such as Cavallo's) attached to a telescope, are highly useful. For measuring small distances, as bases, 50 feet and 100 feet chains, and a portable box of graduated tape, will be necessary. For the purposes of surveying, it is usual to employ a chain 66 feet in length, subdivided into 100 links, each 7*92 inches; the reason for using a chain of this length is, that ten of such square chains are equal to an acre, and therefore the acreage of the several divisions of an estate are found with much greater facility when measured in chains and links, than when the measurements are taken in feet. Eofample 1. In order to find the distance between two trees, A and B (Pig. 9), which could not be directly measured because of a pool which occu- pied much of the intermediate space, I measured the distance of each of them from a third object, C, viz., A C = 588, B C = 672, and then at the point C took the angle ACB between the two trees := 55° 40'. Required their distance. This is an example to Case II. of plane triangles, in which two sides, and the included angle, are given. The work, therefore, may exercise the student: the answer is 593*8. CHiP.V.] HEIOHTS AND DISTANCES. 153 Example 2. Wanting to know the distance between two inaccessible objects, C and D (Fig. 106), wbicb lay in a direct line from the bottom of a tower on whose top I stood, I took the angles of depreuion of the two objects, viz., of the most remote 25 J°, of the nearest 57°. ^4t is the distance between them, the height of the tower A B being 120 feet? HAD = 25' 30', hence BAD = BAH — HAD = 64''30'. flAC=57'' O', hence BAG = BAH — H AC =33'' 0'. Hence the following calculation, by means of the natural tangents. ^OT^ if A B be regarded as radius, B D and B C will be the tangents of tke respective angles BAD, BAC, and CD the difference of ^ose tangents. It is, therefore, equal to the product of the difference »f tie natoral tangents of those angles into the height A B. Thus, nat. tan 64° 30' = 20965436 nat tan 33° = 0*6494076 difference 1-4471360 multiplied by height, 120 gives distance CD = 173*6563200 */ The natural sines, tangents, &c., are easily obtained from ^^^We IV., by subtracting 10 from the characteristics, and then ^iifiing the natural number answering to the logarithms with their ^^^^•neteristics so altered. Example 3. Standing at a measurable distance AB (Fig. 103), on a hori- ^t>l pltne, from the bottom of a tower, I took the angle of eleva- ^ of the top (C) ; it is required from thence to determine the ^l of the tower. In this case there would be given A B and the angle A, to find *C=:ABx tan A. B? logarithms, when the numbers are large, it will be log B C = ^AB + logtanA. ^«.-.If angle J A=ll° 19' then B C = ^ A B very nei » A = 16 42 BC=AAB 5> A = 21 48 BC= 1 AB „ » A = 26 34 B C = ^ A B „ 99 A = 30 58 B C = 1 A B „ >9 A = 35 BC = 375AB 99 A = 38 40 BC= 1 AB „ 99 A = 45 B C = A B, ewacdy. To HYe the time of computation, therefore, the observer may set ^ inttnunent to one of these angles, and advance or recede, till it 154 HEIGHTS AND DI8TAKCSS. [PART I. accords with the angle of elevation of the object; its height above the horizontal level of the observer's eye will at once be known, by taking the appropriate fraction of the distance A B. Example 4. Wanting to know the height of a church steeple, to the bottom of which I could not measure on account of a high wall between me and the church, I fixed upon two stations at the distance of 93 feet from each other, on a horizontal line from the bottom of the steeple, and at each of them took the angle of elevation of the top of the steeple, that is, at the nearest station 5b° 54', at the other 33° 20^. Required the height of the steeple. Recurring to figure 106, we have given the distance C D, and the angles of elevation at C and D. The quickest operation is by means of the natural tansrents, and the theorem A B = -. ® ' cot D — cot C Thus cot D = cot 33° 20' = 1*5204261 C = cot 55 54 = -6770509 Their difference = -8433752 93 "^"•"''^" = i433752 = "«-'''"^'- Eaoample 5. Wishing to know the height of an obelisk standing at the top of a regularly sloping hill, I first measured from its bottom a distance of 36 feet, and there found the angle formed by the inclined plane and a line from the centre of the instrument to the top of the obelisk 41°; but after measuring on downward in the same sloping direction 54 feet farther, I found the angle formed in like manner to be only 23° 45^ What was ttie height of the obelisk, and what the angle made by the sloping ground with the horizon ? The figure being constructed (see fig. 107), there are given in the triangle ACB, all the angles and the side AB, to find BC. It will be obtained by this proportion, as sin C (= 17° 15'= B — A) : AB (= 54) : : sin A(= 23° 45') : BC = 733392. Then, in the triangle D B C are known B C as above, B D = 36, C B D = 41°; to find the other angles, and the side CD. Thus, first, as CB -f BD : CB — BD : : tan|(D -f C) = ^(139°) : tanJ(D — C) = 42° 24 J'. Hence 69° 30' + 42° 24|' =112° 54 J' = C D B, and 69°30' — 42°24|' = 27°5J' = BCD. Then, sin BCD : BD :: sinCBD : CD = 51 '86, height of the obelisk. The angle of inclination DAE = HDA = CDB — 90° = 22° 54i'. Remark. — If the line BD cannot be measured, then the angle D A E of the sloping ground must be taken, as well as the angles C A B and C B D. In that case D A E + 90° will be equal to C D B: HEIGHTS AND DISTANCES. 155 ifter C B is found from the triangle A C B, CD may be the triangle C B D, by means of the relation between the . the sines of their opposite angles. Example 6. ; on a horizontal plane, and wanting to ascertain the height of standing on the top of an inaccessible hill, I took the angle •tion of the top of the hill 40% and of the top of the tower en measuring in a direct )ine 1 80 feet farther from the hill, I 1 the same vertical plane the angle of elevation of the top of wer 33^ 45^. Required from hence the height of the tower. e figure being constructed (see fig. 108), there are given, AB «0, C A B = 33° 45', ACB = CBE — CAE=17°15', CBD r, BDC = 180° — (90° — DBE)=130°. And CD may be ad from the expression C D . rad^ = A B . sin A . sin C B D . cosec :B.8ecDB£. Or, Dsing logarithms. log AB = 180» = 2-255273 + log sin A = 33° 45' = 9-744739 -f log sin CBD = 11° 0' = 9-280599 -f log cosec :ACB = 17° 15' = 10-527914 -f log sec DBE *"~ 40° 0' "~~ 10-115746 41-924271 ] log CD = log rad^ 83-9983 = 40-000000 1 924271 Example 7. In order to determine the distance between two inaccessible ob- KcttAand B (Fig. 108), on a horizontal plane, we measured a con- TenieDt base, C D, of 536 yards, and at the extremities C and D took ^following angles, via., D C B = 40^ 16', B C A = 57" 40', C D A =5 KV 22', A D B = 71° 7'. Required the distance A B. Rnt, in the triangle C D A are given all the angles, and the >^CD to find AD. So, again, in the triangle CDB, are given iQtlietDgles, and CD to find DB. Lastly, in the triangle DAB ^ giren the two sides A D, B D, and the included angle A D B, to fad A B=i 939-52 yards. £mari.~ In like manner the distances taken two and two, be- ^*^ soy number of remote objects posited around a convenient station line, may be ascertained. Example 8. Soppote that in carrying on an extensive survey, tbe distance be- ^ecn two spires A and B (Pig. 109) has been found equal to 6594 1^*^ and that C and D are two eminences conveniently situated for eitending ibe triangles, but not admitting of the determination 156 HEIGHTS i\ND DISTANCB8. [PABT I. of tlicir distance by actual admeasurement: to ascertain it, therefore, we took at C and D the following angles, viz. : — CA C B = 85° 46' JA D C = 31° 48' (B C D = 23° 5G' |a D B = 68° 2' Required C D from these data. In order to solve this problem, construct a similar quadrilateral Acdhy assuming cd equal to 1, 10, or any other convenient number: compute A h from the given angles, according to the method of the preceding example. Then, since the quadrilaterals Acdh^ A C D B, are similar, it will be, as Ah : cd i: AB : Cl>; from which C D is found to be equal to 4694 yards. Example 9. Given the angles of elevation of any distant object, taken at three places in a horizontal right line, which does not pass through the point directly below the object; and the respective distances between the stations; to find the height of the object, and its distance from either station. Let A EC (Fig. 110) be the horizontal plane, FE the perpen- dicular height of the object above that plane, A, B, C, the three places of observation, FAE, FBE, FCE, the angles of elevation, and A B, B C, the given distances. Then, since the triangles A E F, BEF, CEF, are all right angled at E, the distances A£, BE, CE, will manifestly be as the cotangents of the angles of elevation at A, B, and C. Put AB = D, BC = </, EF = a?, and then express algebraically the theorem given in Geom. Sect. III. Prop. XXV., which in this case becomes, AE2.BC-hCE-.AB = BE2.AC-fAC.AB.BC. The resulting equation is dx^^QOiAf 4- Da?^(cotC)- = (D + rf)ar^(cotB)« + (D + d)l}d. From which is readily found (D-f fl?)P<^ rf(cot A)2 + D(cotC)^ - (D + <3?) (cot Bf ' Thus £ F becoming known, the distances A E, BE, C £, are found, by multiplying the cotangents of A, B, and C, respectively, by EF. Remark. — When D=</, or D-f-rf=2D = 2</, that is, when the point B is midway between A and C, the algebraic expres- sion becomes, a? = </-i. >/ 1 (cot A)* -h I (cot cy — (cot By, which is tolerably well suited for logarithmic computation. The rule inay, in that case, be thus expressed. Double the log cotangents of the angles of elevation of the ex- treme stations, find the natural numbers answen'ng thereto, and take half their sum ; from which subtract the natural number answering CHAP, v.] HSIOHTS AND DISTANCES. 157 to twice the log cotangent of the middle angle of elevation : then haiftbelogof this remainder subtracted from the log of the mea- sured distance between the first and second, or the second and third stations, will be the log of the height of the object. The distance from either station \%ill be found as aboye. NtU.-The case explained in this example, is one that is highly useful, and of frequent occurrence. An analogous one is when the angles of elcTation of a remote object are taken from the three angles of a triangle on a horizontal plane, the sides of that triangle being known, or measurable : but the above admits of a simpler computa- tion, and may usually be employed. Example 10. From a convenient station P (Fig. Ill), where could be seen tliree objects, A, B, and C, whose distances from each other were known (ra. A B = 800, A C =-600, B C = 400 yards), I took the horizon- til ingles A P C = 33° 45', B P C = 22° 30'. It is hence required to detemine the respective distances of my station from each object. Here it will be necessary, as preparatory to the computation, to descnbe the manner of 0(Mttnteti(m, — Draw the given triangle ABC from any convenient Kile. From the point A draw a line A D to make with A B an angle eqoal to 22° 30', and from B a line BD to make an angle DBA = 33^45'. Let a circle be described to pass through their intersection D, ud through the points A and B. Through C and D draw a right line to meet the circle again in P: so shall Pbe the point required. For, drawing PA, PB, the angle A PD is evidently = AB D, since it ^ds on the same arc A D : and for a like reason B P D = B A D. So that p is the point where the angles have the assigned value. The result of a careful construction of this kind, upon a good-sized ««lc, will give the values of P A, P C, P B, true to within the 200th pwtofeach. Manner of Computation, — In the triangle ABC, where the sides *n known, find the angles. In the triangle A B D, where all the iiglesare known, and the sides A B, find one of the other sides A D. We B A D from B A C, the remainder, D A C, is the angle included Ween the two known sides, AD, AC; from which the angles ADC ttdACD may be found, by Case II., p. 149. The angle C A P = W - (A P C + A C D). * Also, BCP = BCA — ACD; and 'BC = ABC H- PBA=:ABC-f- sup. ADC. Hence, the ^^ required distances are found by these proportions. As sin APC: AC :: sin PAC : PC :: sin PC A : PA; and lastly, as sin *PC : BC : : sin B P C : B P. The results of the computation are, 'A= 709-33, PC = 1042-66, PB = 934 yards. V The computation of problems of this kind, however, may be 1 litde shortened by means of an analytical investigation. Those ^ wish to pursue this department of trigonometry may consult the •"•*»es by Bmnycattky Gregory ^ and WwdhoxMe, 158 HSIOHTS iIND DISTANCES. [PART I. Noie. — If C had been nearer to P tlian A B, the general principles of constmction and compotation would be the same; and the modi- fication in the process Terr obyioas. DBTSEMINATION OF HSIOHTS AND DISTANCES BY APPBOXIMATS ME- CHANICAL METHODS. 1. For HeigkU. 1 . By sAadatPS^ when the son shines. — Set op yertically a staff of known length, and measure the length of its shadow upon a horizon- tal or other plane ; measure also the length of the shadow of the ob- ject whose height is required. Then it w\]\ be, as the length of the shadow of the staff, is to the length of the staff itself; so is the length of the shadow of the object, to the object's height. 2. By two rods or staves set up vertically: — Let two staves, one, say, of 6 feet, the other of 4 feet long, be placed upon horizontal circular or square feet, on which each may stand steadily. Let A B (Fig. 112) be the object, as a tower or steeple, whose altitude is required, and A C the horizontal plane passing through its base. Let C D and E F, the two rods, be placed with their bases in one and the same line C A, passing through A the foot of the object; and let tliem be moved nearer to, or farther from, each other, until the summit B of the object is seen, in the same line as D and F, the tops of the rods. Then bv the principle of similar triangles, it will be, as D H (= C E) : F H : : D G (= C A) : BG ; to which add A G = C D, for the whole height A B. 3. By reflection, — Place a vessel of water upon the ground, and recede from it, until you see the top of the object reflected from the smooth surface of the liquid. Then, since by a principle in optics, the angles of incidence and reflection are equal, it will be as your distance measured horizontally from the point at which the reflection is made, is to the height of your eye above the reflecting surface ; so is the horizontal distance of the foot of the object from the vessel to its altitude above the said surface *. 4. By means of a portable barometer and thermometer, — Observe the altitude B, of the mercurial column, in inches, tenths, and hun- dredths, at the bottom of the hill, or other object whose altitude is required ; observe, also, the altitude, 6, of the mercurial column at the top of the object ; observe the temperatures on Fahrenheit's thermo- meter, at the times of the two barometrical observations, and take the mean between them. * Leonard Digge§, in his curious work, the PatUometria^ published in 1571 « first proposed a method for the determination of altitudes by means of a geo- metrical square and plummet, which has been described by various later authors, as Ozanam, Donn, Hutton, &,c. But, as it does not seem preferable to the methods above given, I have not repeated it here. r.] HEIGHTS AND DISTANCES. 159 B — i 55000 X = height of the hill, in feet, for the tempera- of SS"" on Fahrenheit. Add ^ Jg of this result for every degree d) the mean temperature exceeds 55°; suhtract as much for every ree below 55°. rhis will he a good approximation when the height of the hill is « dian 2000 feet ; and it is easily remembered, because 55°^ the flomed temperature, agree with 55, the significant figures in the co- fident; while the significant figures in the denominator of the cor- .•sctiDg fraction are two /burs, 5. Bifan extension of the principle of page 153.— Set the sextant, »r other instrument, to the angle 45°, and find the point C (Fig. 103) >zi the horizontal plane, where the object A B has that elevation : then set ibc bstrument to 26® 34', and recede from C, in direction BCD, kill the object has that elevation. 7% Munce C D between the two stations will ^ = A B. So,sgain, if C = 40%D = 24°3lJ', CD will be = A B. or, „ if C = 35% D = 22° 23', CD „ = A B. or, „ if C = 30°, D = 20° 6', C D „ = A B. or, „ if C = 20°, D = 14° 5e\ CD „ = A B. or, generally, if cot D — cot C = rad, CD „ = A B. ^- For deciation from lend. — Let E represent the elevation of the ^^i>^t line to the earth above the true level, in feet and parts of a '^K D the distance in miles: then E = | D'^ Tbis gives 8 inches for a distance of one mile ; and is a near ap- P*^ximauoD when the distance does not exceed 2 or 3 miles. 2. For Distances, 1* By means of a rhombus set off upon a horizontal plane, — Sup- P^O(Fig. 113) the object and O B the required distance. With ^ ibe or measuring tape, whose length is equal to the side of the in- ^M rhombus, say 50 or 1 00 feet, lay down one side B A in the direction BO towards the object, and BC another side in any con- '^taJeni direction (for whether B be a right angle, or not, is of no ^^'''^(tliience) ; and put up rods or arrows at A and C. Then fasten ^ ends of two such lines at A and C, and extend them until the two other ends just meet together at D ; let them lie thus stretched *pOD the ground, and they will form the two other sides of the "•^^inbas AD, CD. Fix a mark or arrow at R, directly between C ^ 0, upon the line A D ; and measure R D, R A upon the tape. '^ it will be as R D : D C : : C B : B O, the required distance. OdksTwise, To find the length of the inaccessible line Q R. At some convenient point B (Fig. 114) lay down the rhombus J^ADC, 10 that two of its sides, B A, B C, are' directed to the ex- ^''"Wwi of the line Q R. Mark the intersections, O and P, of A R, 160 HEIGHTS AND DISTANCES. [PART I. C Q, with the sides of the rhomhus (as in the former method) : then the triangle O D P will he similar to the triangle R B Q ; and the in> accessihle distance R Q will be found = — -.* OD X DP Thus,ifBA =BC,&c. =100fk.,OD = 9 ft. 5 ins.,D P = 11 ft. 10ins.,OP = 13 ft. 7 ins., then QR = ^^t^-^^-~^ = 1219 feet. 2. By means of a micrometer attached to a telescope. — Portable in- struments for the purpose of measuring extremely small angles, hare been invented by Martin, Cavallo, Dollond, Brewster, and others. In employing them for the determination of distances, all that is neces- sary in practice is to measure the angle subtended by an object of known dimensions, placed either vertically or horizontally, at the re- moter extremity of the line whose length we wish to ascertain. Thus, if there be a house, or other erection, built with bricks, of the usual size ; then four courses in height are equal to a fooi^ and four in length equal to a yard: and distances measured by means of these will be tolerably accurate, if care be taken while observing the angle subtended by the horizontal object, to stand directly in front of it. A man, a carriage-wheel, a window, a door, &c., at the remoter ex- tremity of the distance we \^ ish to ascertain, may serve for an ap- proximation. But in all cases where it is possible, let a foot, a yard, or a six-feet measure, be placed vertically, at one end of the line to be measured, while the observer with his micrometer stands at the other. Then, if h be the height of the object, either | A x cot \ angle subtended or A X cot angle will give the distance, according as the eye of the observer is hori- zontally opposite to the middle^ or to one extremity of the object whose angle is taken. When a table of natural tangents is not at hand, a very near ap- proximation for all angles less than half a degree^ and a tolerably near one up to angles of a degree^ will be furnished by the following rules. If the distant object whose angle is taken be 1 foot in length, then 3437*73 -6- the angle in minutes \ will give the distance in or 206264- -^ the angle in seconds i feet. If the remote object be 3, 6, 9, &c., feet in length, multiply the former result by 3, 6, 9, &c., respectively. Ex. 1. What is the distance of a man 6 feet high, when he sub- tends an angle of 30 seconds ? • ForPD: DA:: AB: BR = ^^ ; andOD:OP::BR!RQ = A B' . O P O D . D P* CHAP, v.] HBI0HT8 AND DISTANCES. 161 206264 X 6 -r 30 = 206264 -r 5 = 412528 feet =t= 137509 yards, the distance required. Ex, 2. In order to ascertain the length of a street, I pat up a foot measure at one end of it, and standing at the other found that mea- sure to subtend an angle of 2 minutes : required the length of the street. 3437-73 -r 2 = 1718-86 feet = 57295 yards. 3. By means of the velocity of sound. — Let a gun be fired at the remoter extremity of the required distance, and obserre, by means of a chronometer that measures tenths of seconds, the interval that elapses between the flash and the report : then estimate the distance for one second by the following rule, and multiply that distance by the obserred int^val of time ; the product will give the whole dis- tance required. At the temperature of freezing, 32% the Telocity of sound is 1100 feet per second ; for lower temperatures deduct, or for higher tem- peratures add, half a foot per second for every degree of difference from 32^ on Fahr. thermometer ; the result will show the velocity of sound, very nearly, at all such temperatures. Thus, at the temperature of 50% the velocity of sound is, 1100 X i (50 — 32) = 1109 feet. At temperature 60% it is 1100 + i (60 — 32) = 1114 feet. For a more accurate method of determining the velocity of sound, I page 281. 162 CONIC SECTIONS. [part I. CHAP. VI. CONIC SECTIONS. Sect. I. Dejiniti&ns* 1 . Conic /Sections are the figares made by a plane cutting a 6one. 2. According to the different positions of the cutting plane there arise five different figures or sections, viz., a triangUy a eirde^ an ellipse^ an hyperbola, and a parabola: of vt^hich the three last aie peculiarly called Conic Sections, 3. If the cutting plaoe pass throngh the vertex of the cone, and any part of the base, the section will be a iHan^y as AB. (Fig. 115.) 4. If the plane cut the cone parallel to the base, or make no angle with it, the section will be a circle , as A B. (Fig. 116.) 5. The section is an ellipse when the cone is cut obliqnely through both sides, or when the plane is inclined to the base in a less angle than the side of the cone is, as A B C D. (Fig. 120.) 6. The section is a parabola when the cone is cat by a plane parallel to the side, or when the cutting plane and the side of the cone make equal angles with the base, as P^ A P^ . (Fig. 138.) 7. The section is an hyperbola when the cutting plane makes a greater angle with the base than the side of the cone makes, as PjAP,. (Fig. 130.) iVb/e.— In all the above definitions the cone is supposed to be a right cone. 8. If all the sides of the cone be continued through the vertex, forming an opposite equal cone, and the plane be also continued to cut the opposite cone, this latter section is called the opposite hyper^ bola to the former, as S^ B S, . (Fig. 130.) 9. And if there be two other cones, with their axes in the same plane and their sides touching the sides of the former cones, then will the same plane cut all the cones and form four hyperbolas, as P,AP«, R, CRo, S^BS^, Qj DQ^ (fig. 118), each opposite pair of which are similar ; these hyperbolas are said to be eonjtigate to each other, and the figure thus formed is called the figure of ike conjugaJte hyperbdas, as fig. 118. ^(rfe.— In the following definitions, the letters refer to fig. 117 for the ellipse; fig. 118 for the hyperbola; and fig. 119 for the parabola. CHAP. VI.] CONIO 8I0TI0N8. 103 10. The veriice9 of any section are the points where the cutting pkne meets the opposite sides of the cone, as A and B. Cor, Hence the ellipse and the opposite hyperbolas hare each two vertices; but the parabola only one; unless we consider the other as at an infinite distance. 1 1. The mqfor amt^ or iransverte diameter of a conic section, is the line or distance A B between the Yertioes. Cor. Hence the axis of a parabola is infinite in length, A B being ody a pari of it. 1 2. The eenhre O is the middle of the axis. Cor, Hence the centre of a parabola is infinitely distant from the vertex. And of an ellipse, the axis and centre lie within the curve : bot of an hyperbola, without. Id. The mtHor^ or conjugate aariSj is the line C D, drawn through the centre perpendicular to the transverse axis, and bounded each Way by the curre. CJcr, Hence the parabola has not any conjnsate axis, unless we aappoee it at an infinite distance, and infinite in length. 14. A Umgemi to a curve at any point, is a line as A^ T^ , which tooehes the carve in that point; but being produced either way docs not cut it. 15. A diameter is any right line, as A^Bj, drawn through the csentre, and terminated on each side by the curve ; and the extremi- ties of the diameter, or its intersections with the curve, as A ^ and H J , are its vertices. Cor, Hence all the diameters of a parabola are parallel to the Axia, and infinite in length. Hence, also, every diameter of the ellipse and hjrperbola has two vertices; but of the parabola, only one; aaleaa we consider the other as at an infinite distance. 1^. The coi^ugaU to any diameter is the line drawn throngh the centre, and pandlel to the tansent of the curve at the vertex of the diameitf. So, 0, D, , panllel to the tangent at A ,, is the conjugate toAjBji. 17. An mrdimate to aay diameter is a line parallel to its conjugate, or to tlie tangent at its vertex, and terminated by the diameter and carve. So^ K I and K, Aj are erdinates to the axis A B ; and K, I •ad fi| I| ordiaates to the diameter A, B^ . Cor. ilesoe the ordinatea of the axes are perpendicular to it; but of other diameters, the ordinates are oblique to them. 18. An oiirisfa is a pvt of any diameter, contained between its vertex and an ordinate to it; as AK or BK, and A, E, or B, S, . Cor. Hence, in the ellipse and hyperbola, every ordinate has two abec ioom ; \ml in the parabola only one; the other vertex of the diaeieter beiag infinitely distant. 19. The ^wrtex from which the abcissa are measured is called the origin of those abcissse, and any abcissa and its ordinate are called co^frdinaieSi as AK and KI. 20. The parameter of any diameter is that double ordinate which is a third proportional to the transverse and conjugate axes in the M 2 164 CONIC 8BCTI0N8. [PART I. ellipse or hyperbola, and to any abscissa and its ordinate in the para- bola. The parameter of the transyerse axis is called the principal para- meter^ or the lattis rectum; thus, if A^ B. : C, D, :: Cj D, : I^ I^, then Ig I4 is i\\e parameter of A ^ Bj ; and if AB : CD :: CD : I'j I3, then I J I3 is the lotus rectum, 21. The point where the parameter cuts the transyerse axis is called \he focuB^ as E, F; and the distance of the focus from the nearest yertex of the same axis is called the focal digtance^ as AE, BF. Cor, Hence, the elHpse and hyperbola haye each two foci, but the parabola only one. The foci (or burning points) were so called because all rays are united or reflected into one of them, which pro- ceed from the other focus, and are reflected from the curye. 22. The point £ is also frequently called the poie; and a line drawn from any point in the curye to the pole, is called the radius vector y as E I. And the angle B £ I, contained between the radius yector and the transyerse axis, is called the traced angle, 23. The directrix is a right line drawn perpendicular to the trans- yerse axis of a conic section, through an assignable point in the prolongation of that axis; such that lines drawn from any points in the curye parallel to the axis to meet the directrix, shall be to lines drawn from the same points to the focus, in a constant ratio for the same curye. Thus, if A E : A T :: £ I^ : I«y :: E I3 : I3 « :: E I : I X, then X Y is the directrix. In the ellipse A £ is less than A T. In the parabola A E is equal to A T. In the hyperbola A E is greater than A T. 24. The subtangent to any point in a curye, is that portion of the transyerse axis which is contained between the tangent and ordinate to the same point, as K^ T, . 25. A normal at any point is a line drawn from that point per- pendicular to the tangent, to meet the transyerse axis, as A ^ L. 20. The Radius q/* Curvature of a conic section or other curye, is the radius of that circle which is precisely of the same cunratnre as the curye itself, at any assigned point, or the radius of the circle which fits the curye and coincides with it, at a small distance on each side of the point of contact. The circle itself is called the oseu- latorg circle, or the eguicurve circle ; and if the curye be of inces- santly yarying curyature, each point has a distinct eqaicarye circle, the radius of which is perpendicular to the tangent at the point of contact. 27> An asymptote is a right line towards which a certain cunre line approaches continually nearer and nearer, yet so as never to meet, except both be produced indefinitely. The hjrperbola has two • asymptotes, as UV, WZ. CHir. VI.] CONIC SBCTIONS: ELLIPSE, 165 Sbct. II. Properties of the Ellipse. 1. Let the frostum of a right cone AGBH (fig. 120), he cut by apkne fonning a tangent to each end, then will the section (ACBD) thus prodaced be an ellipse. Let dszQBy the diameter of the lesser end; D = AH the diameter of the greater; A = AG, or BH, the dant height; t=iAB^ the transverse diameter of the ellipse ; c = C D, the eonjugile diameter; andysEF, the distance between the two foci; then, ^=A2 + Drf (1.) c» = D^ (2.) / =A (3.) 1 Lety=IK (fig. 117), any ordinate; a;=:AK, its abscissa nettored from A ; ;y =: O K, its abscissa measured from the centre, O;0 = IE, the radiup vector from the focus £; i9= the traced »a^ IBF; and ^= >^i^'— i<^; then, y = f ^(/4r-««) (4.) viiich b the equation to the ellipse when the absciss® are measured ftm the vertex ; y=f Va<'-**) (5.) vbich is the equation when the abscissas are measured from the •tttre; and ^^ it — g.coBS ^ '^ thich is called the polar equation^ and finds its principal use in the miei^mtions of astronomy. 3. For the relation between the abscissse and ordinates in any •%ie, we have. As the square of the transverse azis^ : the square of the conjugate ; : : the rectangle of the abscissas, : the square of their ordinate. Or, /«:tf^::«(/-«):y» (7.) "Rw sune proportion obtains between the ordinate and abscissae to •^ditmetcr; or putting «, for Aj^j (fig. 117); w^ for B^Kj, ^^y,forIKi; ^^^ t,*:e,^::w,x^:y,* (8.) ^ tarn of the squares of any pair of conjugate diameters is ^ to the sum of the squares of the two axes; or putting t^ for ^i^ (fig. 117), and e^ for C^ D^ ; then t^ + e'^t^^^e,^ (9.) ^U the parallelograms that can be circumscribed about an ellipse •reequil to the rectangle of the two axes; or = ct (10.) 106 CONIO 8BCTI0N8: BLL1P8B. [pABT I. The sum of the two radius vectors, drawn from the two foci, to any point in the curve, are equal to the transverse axis; or putting e = £ I (fig. 117), the radius vector from £, and V = F I, the same from F ; we have V -hr = / (11.) Cor. The distance of either focus from either extremiljr of the conjugate diameter is ^oal to half the tmnsverae axis ; or, CE =CF = | (12.) The square of the distance between the two foci equals the differ- ence of the squares of the two axes ; or, r^t'^e' (1«.) 4. tf TM (fig. 121) be the tangent to an ellipse at any point I, and let T be the point in which the tangent meets the transverse axis produced; also from the two foci draw FM, EN, perpendicular to the tangent, and produce E N and F I to meet each other in P ; then, OK : AO :: AO : OT; or, OT — -- (14.) Cor. If there be any number of ellipses described on the same transverse axis (fig. 122), and any ordinate be drawn, cutting all the ellipses in the points I^, I^, I^, I^, &c., the tangents to the several ellipses at those points, wilt all meet the transverse axis produced in one common point, T 0^') The angles made by tiie two radius vectors and the tangent arc equal; or, 2LTIE=:/.LIF (16.) The points M and N (fig. 121) fall in the circumference of a circle Ivhose diameter is the transverse axis, AB (17.) Also, EN. FMarCO'tfc ^1 J (18.) And, EN* = (I)'t (•••) If m be in the middle of K I , , then, A m prodctced wOi meet the two tangents TQ, BQ, in their point of intersection, Q ... (20.) If the ordinate 1 1 ^ passes through the focus, tben the point T of intersection of the tangent and the transverse axis prodac^ will be a point in the directrix (21*) c' In any ellipse the parameter = — (^^0 5. Let R ss the radius of curvature at any point in an ellipse, and V, «, the distances of the same point from the two foci; then R = *^ (23-> CBAf. YI.] COMIC MCTIOVS: BLLIP81. 167 It is A mtunmum at the extremity of the conjugate axis, when *^-f. ("•> And a minimum at the extremity of the transverse axis, when »* = ^ (25.) The area of any ellipse is a mean proportional hetween the areas of the circles deambed on its two axes (^^0 P90BLSM8 jaiLATINO TO THX ELLIPSE. ^«OB. I. To find the two foci E^ Fy of an ellipse^ of which the transverse cune A B, and conjugaU C D, are given. Prom C or D AS a centre, with a radius equal to the semi- trans- ^'cvce axis AC or BO, describe an arc cutting the transverse axis in ^ J, wLich are the two foci required. (Fig. 123.) FiOB. II. To von^mct an ellipse whose two axes are given, Knd the distance £F, from Prob. I., and let a fine thread, £IF£, ?* length = £F + A B, be put round two pins fixed at the points ^ ^; theo, if a pencil be put within the cord, and carried round, ^^>^ being taken diat the cord is alwavs tight, the pencil will dc- •*^*"ibe an ellipse, A I C B D A. (Fig. 1 24*) ^^^^41. III. The transverse and conjugate amis of an ellipse being given^ to describe the same with a trammel, let the distance between the pencil A and the first pin B, be ^^^ to half the conjugate axis, and the distance between A and ^^^ leeond pin C, be equal to half the transverse ; then, the pins ****iBg pot into the grooves of the trammel, and the pencil A being **^^>^ will describe the ellipse. (Fig. 125.) OQierwise, — ^Let there be provided three mlers, of which the two ^^) FH, are of the same length as the transverse axis AB, and the ^^vdHP, equal in length to £F, the focal distance. Then, con- ^^^cting these rulers so as to move freely about £ F H and I^ their ^•iimtiun I will always be in the curve of the ellipse : so that, ^f tkere be alits mnning along the two rulers, and the apparatus ^*ned fiteiy aboat the foci^ a pencil put through the slits at their ^^ sf intenectaon will deacribe the carve. (Fig. 126.) Pbob. ly. To find the two axes of any given ellipse, ^W tny two pamUel lines across the ellipse, as M L, F K : bisect "•"» in the points I and D, through which draw the right line 168 CONIC sections: hypbrbola. [part i. N I E P, and bisect it in O. From O as a centre, with any adequate radius, describe an arc of a circle cutting the ellipse in the points G, H. Join O, H, and parallel to the line G, H, draw through O the minor axis CD; perpendicular to which through O draw AB, which will be the major axis. (Fig. 127.) pROB. V. From any given point out of an ellipse to draw a tangent to it. Let T be the given point, through it and the centre C draw the diameter A B ; and parallel to it any line H I terminated by the curve. Bisect HI in O ; and C O produced will be the conjugate to A B. Draw any line TS = TB, and make TR = TC. Draw RA, and parallel to it, SP cutting AB in P. Through P, draw PM parallel to CD, and join TM, which will be the tangent required. (Fig. 128.) Prob. VI. To find the length of an elliptic arclj C I^ . Produce the height CE to O, the centre of the ellipse. Join I J and O, and from O as a centre with radius C O eqttai the semi- conjugate, describe an arc cutting I , O in a. Bisect 1 1 a in 5, and from O as a centre with radius O b describe the arc bcy catting C O produced in c; then the circular arc ^c is equal in length to half the elliptic arc I, C Ig . (Fig, 129.) Sect. III. Properties of the Hyperbola. 1. Let two opposite right cones (Fig. 130) be cut by a plane making a less angle with their axes than the sides of the cones make, then the sections thus produced will be two opposite hyperbolas. Let d = AHy the lesser diameter of the cone at the vertex of one hyper- bola ; D = B G, the greater diameter of the cone at the vertex of the opposite hyperbola ; /t = A G, or B H, the slant height ; / = A B, the transverse axis; c = CD, the conjugate axis; and /*= £F, the dis^ tance between the two foci ; then we have /«=:A2-Drf . . . (1.) c' = Drf . . . . (2.) f = h . . . . (3.) which will be seen to be identical with the corresponding equations for the ellipse, with the exception of the first, which, however, only differs in the sign of the last term. This at once indicates a generu analogy between the properties of the two curves ; and if we employ the same letters to represent the corresponding lines in figures llTi 121, 122, and figures 118, 131, 132, the formula already given for the ellipse will apply to the hyperbola, only making the following alterations. CHAP. VI.] 4x>Nic sections: hypkbbola. 169 (4.) Becomes y = T/V/ ^'* "'' *'^' (5) « 9=-fAji''-\n- (6.) „ f> = ^ I < + ^ • COS /3 (7.) and (8.) remain unaltered. (9.) Becomes <,* — c^' = f* — c*. (10.) All the parallelograms that can be inscribed between four conjugate bjrperbolaSy are equal to the rectangle of the two axes ; or, = et (11.) Becomes V— © = ^ (12.) „ OEorOF = ACorCB. (13.) „ r^f-^e. (14.) to (25.), both inclusiTe, remain unaltered. 2. Besides these, however, there are several curious properties which relate to the tuymptotes of the hjrperbola. Let 8), '3, 829 (fig. 133) be an hyperbola; OV and OZ its asymptotes; 9^ K«^, a double ordinate to the transverse axis; O^r^, Oz^^ Oz^^ Oz^^ Oj»^, abscissa taken on the asymptote OZ; and;!;, 'i>^8 '^^ ^3 'si ^4 '49 z^ «5, ordinates to the same, drawn parallel to the other asymptote OV: Then the parallelograms 0©^ *, ^r,, Ov^ 9^ z^y O©,, «, z^y &c. ire all equal ; or 0*1 -^1 *i = ^^« • *2 *s =0^3 .^3 *j, =04r4 .«4«4 (26.) /« Also, m#i . <i *a = ««5 . «i 85 = ^*s' = J • • (27.) And the triangle O T I, (Fig. 131) = the triangle O B Q . (28.) Also, if the abscissae Oz^^Oz^^Oz^^ &c. of any h^rperbola, be taken on one of the asymptotes in an increasing eeometncal progres- sion, the ordinates z^9^y z^ ^^ , z^ 9^^ &c. parallel to the other asymp- tote are in decreasing geometrical progression, having the same imtio (29.) And, when the distances Oz^y Oz^^ &c. are in geometrical pro- gression, the asymptotic spaces z^ 9. So z^^ Zy «. 9^ jT,, &c. will be m arithmetical progression, and will, therefore, be analogous to the logarithms of the former (30.) The radix of the system of logarithms will depend upon the value of the angle made by the two asymptotes. In Napier 9 logarithms V O Z is a right angle ; while in the common logarithms it is 24** 44' 254" ♦. * See Sect. 6, page 93. 170 COXIC 8BCTI0K8: HYFIRBOLA. [PABT I. 3. In the cmse of foar conjngate bypcrboIa8, the transyerse and con- jugate axes of one pair become the conjugate and transTerse of the other pair, and th^efore they are said to be matually conjugate to each other. PBOBLRMS RELATING TO HYPERBOLAS. Prob. I. To detcrihe an Hyperbola of tckich the tranwerM axis and twofod (ire given. Let one end of a long ruler, £0 (fig. 134) be fictttened at one of the foci, £, so as to turn freely about that point as a centre. Then take a thread F I , O, of such a length, that when one end is fixed at the other focus F, and the other end to the ruler at O, the doubled part will just reach to B, one end of the transverse axis. Then if the ruler E O be turned about the fixed point E, at the same time keep- ing the thread O I , F always tight, and its paK I , O close to the side of tlie ruler, by means of the pencil I j ; the curve line B S . de- scribed by the motion of the pencil I, is one part of an hyperbola. And if the ruler be turned, and move on the other side of the fixed point F, the other part B S^ of the same hyperbola may be described after the same manner. But if the end of the ruler be fixed in F, and that of the thread in £, the opposite hjrperbola P ^ A P, jao^y be described. Othencise; also by continued motion. Let E and F (Fig. 135) be the two foci, and A and B the two vertices of the hyperbola. Take three rulers, C D, D I, I F, and fix them so that E D s: G F ax A B, and DC =: EF; the rulers D I and 6 1 being of an indefinite length beyond I, and having slits in them for a pin to move in ; and the rulers having holes in them at £ and F, to fiasten them to the foci £ and F by means of pins, and at the points D and C they are to be joined by the ruler D C. Then, if a pencil be put in the slits, at the common intersection of the rulers D I and F I, and moved along, causing the two rulers F I, ID, to turn about the foci E and F, that pencil will describe the portion A P j of an hyperbola. Prob. II. To describe an hyperbola of which the two axes aregiveuy by finding points in the curve. Let A B be tlie transverse axis, and C D the conjugate. First from O as a centre, with radius equal A C or C B, describe arcs cutting the tuansverse axis produced in £ and F, which will be the two foci. Then assume any number of points, a^t a,, a^y &c. beyond F, and from £ as a centre, with radii equal to Ba,, Bao, Ba,, &c., describe arcs ^,9 ^^9 ^89 ^^* ' ^^^ ^^^^ F as a centre, with radii equal to Aa,, Aa.„ Aa^, &c., describe arcs cutting the former arcs in the points c^^c^^ C3, &c., which will all be points in the hyperbola re- quired. (Fig. 136.) CHAP. VI.] CONIC BJraTlONB: PABABOLA. 171 Prob. III. To deteriht an kyperhokL, of which the transvene axis A B, and any abscissa A O, and its doubU ordinate H I, are given. Through A draw a h parallel to H I ; from H and I draw H a and I h parallel to A O ; then divide aH and &I, each into any number of equal parta, 1, 2, 3, &c., and through the points of division draw lines radiating to A as a centre ; also divide K H and K I into the same number of equal parts, and through the points of division draw lines nuliatiog to B, then will the several points «,, Cg, C3, &c, in which these lines intersect the former lines, be so many points in the hyperbola required. (Fig. 137.) Sbct. IV. Properties of the Parabola. 1. If the right cone KHI (fig. 138) be cut by a plane parallel to the side KH, then will the section P^ A P,, thus produced, be a parabola. Let the cone also be cut by a plane, M A, perpendicular to the plane ^ the parabola, and by another plane, G A, jiarailel to the base of the cone, and both passing through the point A ; bisect the distance M O (between their two vertices) in L, and through L, take a plane L T, parallel to the base of the cone, and let S be the focus of the parabola. Let & = B P^, half the base of the parabola ; d = OA, the dinneter of the cone at the vertex of the paral)ola ; /= A E, ike focal diBtanoe ; and A s K A, the alant height of the cone above dieaame; iImo The Mm TX, in which the plane L T meets the plane of the para- bola, is its directrix ... ... (1.) d^ TA=/=lj (2.) b 2 f-Vd w 2 (M O) = 4/ = the parameter . . (4.) 2. Let /> = I^ I3 (fig. 119), the parameter of a parabola; « vs A &, aay aheeisaa ; jf bb I, X, the corresponding ordinate ; v = £ I, the radiiB vector; and /9 ss the tnced angle II B, £ being the focus; then f^px. . . . (5.) f> = , ^^ ^ . . (6.) 1 q: cos g ^ ' the ayiitiopg to the paiabok : im the latter of which, or the polar eauation, the sign + obtains when K is between A and S, and — when K is below £. (4^ -f p\\ Rad. of curvature at I = „ ^^^ . (7.) 2 >/;t> ^ 172 CONIC sections: parabola. [part i. At the vertex, A, x vanishes, and we have rad. of curv. at vertex = i /> • • . (8.) 3. In the same figure, where XY is the directrix, the following properties ohtain ; viz. AE = AT, EIj = Igy, EI3 =130:, EI =IX, &c.(9.) AsAK : AK, :: KI* : K.P 2; or ^^ = ^^ ^ v (10.) where A K and A R, are any ahscissss, and K I, K3 P ^ their corre- sponding ordinates. Also EI = AK + AE, EPj = AK3 4- AE . (11.) AE = ni=Lil (12.) Ij I3 heing the parameter = p, As/?:K,P^ + Kgl :: K,Pj - K,I :AK3 - AK, ^ . . or, asjt? : P^a :: aP^ : KgK3 i 4. Again, let A , T ^ he the tangent to a parahola at any point A|, and let B ^ A J Y he drawn through A . parallel to the axis A B ; let A J L he perpendicular to A^ T. ; then is K, T, the suhtangent, A . L the normal, K^ L the suhnormaJ; and the following properties obtain; viz. angle EA, T^ = angle ET, A^ = angle T,Aj Y . . (15.) angle LAjBj = angle LA^E (1^0 EA,= ETi (17.) AKj = ATj (18.) subtangent K, Tj = 2 AKg (19.) V subnonnal K, L = 2AE = ^a constant quantity . • (20.) 5. In figure 139 also, where CQ is a tangent to the parabola at the point C, and IK, OM, QL, &c., parallel U) the axis AD. Then IE : EK :: CK : KL . . (21.) and a similar property obtains, whether CL be perpendicular or ob- lique to TD. The external parts of the parallels IE, TA, ON, QL, &c., are always proportional to the squares of the intercepted parts of the tan- gent; that is, the external parts IE, TA, ON, QL, \ are proportional to CI', CT', CO% CQ^ [ . . (22.) or to the squares CK^ CD^, CMV CL', ' CHAP. VI.] CONIC sections: pababola. 173 And as this property is common to every position of the tangent, if the lines IE, T A, ON, &c., he appended to the points I, T, O, &c., of the tangent, and movahle ahout them, so as always to hang verti- cally, and of such lengths that their extremities E, A, N, &c., he in the curve of a parahola in any one position of the tangent ; then making the tangent revolve ahout the point C, the extremities E, A, N, &c., will always form the curve of some parahola, in every position of the tangent. The same properties, too, that have heen shown of the axis, and its ahscisses and ordinates, &c., are true of those of any other dia- meter. PROBLEMS RELATING TO THE PARABOLA. Pbob. I. To anutruet a Parahola^ of which the base and height are given. Construct an isosceles triangle ABD (fig. 140) whose hase AB shall he the same as that of the proposed parahola, and its altitude CD twice the altitude G V of the parahola. Divide each side AD, DB, into 10, 12, 16, or 20, equal parts [16 is a good numher, hecause it can he obtained by continual bisections], and suppose them num- bered 1, 2, 3, &c. from A to D, and 1, 2, 3, &c. from D to B. Then draw right lines I, 1; 2, 2; 3, 3; 4, 4; &c., and their mutual inter- section will beautifully approximate to the curve of the parahola AVB. Pbob. II. To describe a Parabola by finding points in the curve^ an ab- scissa A B, and its double ordinate C D, being given. Through A (fig. 141) draw a b parallel to CD ; and through C and D draw aC, 6D parallel to AB; then divide aC and bD into any convenient number of parts, 1,2, 3, &c., and through these divisions draw lines radiating to A ; also divide B C and B D into the same number of parts, and through the divisions draw lines perpendicular to CD, then the points c,, c^* c^, &c., in which these lines intersect the former, will be in the parahola. Let the ruler, or directrix BC (fig. 142) be laid upon a plane with the square ODO, in such a manner that one of its sides DO lies along the edge of that ruler; and if the thread F M O, equal in length to D O, (Uie other side of the square,) have one end fixed in the extre- mity of the ruler at O, and the other end in the focus F : then slide the side of the square D O along the ruler B C, and at the same time keep the thread continually tight by means of the pencil M, with its part MO close to the side of the square DO; so shall the curve A M X, which the pencil describes by this motion, be one part of a parabola. And if the square be turned over, and moved on the other side of the fixed point F, the other part of the Rame parabola A M Z will be described. 174 CONIC 8BCTI0N8 : PARABOLA. [PART I. Pbob. III. Any right line being given in a parabola^ to find the corre- sponding diameter; aleoy the axis, parameter^ and focus. Draw HI (fig. 143) parallel to the given line DE. Bisect D £, and H I, in # and O, through which draw A O O for the diameter. Draw H R perpendicular to A O and hisect it in B; and draw V B parallel to A O for the axis. Make V B : H B : : H B : parameter of the axis. Then ^ the parameter set from V to F gives the focus. Prob. IV. To draw a tangent to a Parabola, If the point of contact C (fig. 144) he given, draw the ordinate, CB, and produce the axis until AT = A B : then join T C, which will he the tangent. Or if the point be given in the axis produced: take A B ^ AT, and draw the ordinate B C, which will give C the point of contact ; to which draw the line T C as before. If D be any other point, neither in the cunre nor in the axis pro- duced, through which the tangent is to pass: draw DE O perpendi- cular to the axis, and take D H a mean proportional between DE and DO, and draw H C parallel to the axis; so shall G be the point of contact, through which and the given point D the taD|*mit D C T is to be drawn. When the tangent is to make a given angle with the ordinate at tbe point of contact : take the absciss A I equal to half the parameter, or to double the focal distance, and draw the ordinate I £ : also draw A H to make with A I the angle A H I equal to the given angle; then draw H C parallel to the axis, and it will cut the curve in C the point of contact, where a line drawn to make the given angle with C B will be the tangent required. CHAF. VII.] CURVES. 175 CHAP. VII. PROPBBTIER OP CURVES. Sect. I. Definitions. # 1. A curve line^ as already defined, (Def. 5, page 98,) ia a line which continually changes its direction. 2. Apiane carve, is one the several points of which are all situate in the Mme plane ; and a curve of double curvature is one whose several points are not aJl in the same plane, and which, therefore, curves in two directions. 3. Curve lines have heen further divided into AlgebraiccU or Geo- metrieat^ and TrtxMtemiental or MeckanicaL 4. An tdgehrauud curve, is one in which the relation which the abseista bears to the ordinate, taken for any possible point in that corye^ may always be expressed by a constant algebraical formula. 5. A transcendental cuTvCy is one in which no such constant rela- tion between its abscissse and ordinates exists, and which can, there- fore, only be described by mechanical means. 6. The algebraical formula which expresses this relation is called the equation of the curve; and curves are classed into orders, accord- ing to the number of dimensions contained in such equations. Thus any curve line which might be represented by the equation a: a + ^x -H cy + da^ + exy +/y*, would be called a line of the second order, because the abscissa {ai) and ordinate (y) are involved only to the second power; and a curve which might be expressed by = • + 6« + cy -h <^^ -h ««y +ff -h gas" + hw-y + hf» + If would be called a line of the third order, because the same quantities are involved to the third power. The letters a, &, c, &c., are merely the coefficients depending upon the particular curve expressed by the equation. 7. An am/mpMe to a curve is a straight line, to which the curve line continually approaches, without ever meeting it; as the asymp* iotes U V and W Z, to the hvperbola, fig. 118. 8. If a curve cuts itself by passing twice through the same point, as A, fig. 145, this point is called &punctum duplea: if three times, as K, it is called a punctum tripUx^ &c. 170 curves: conchoid. [part i. 9. The oval contained between A and C, fig, 145, is called a nodus: when the distance between A and C becomes indefinitely small, so that the nodus vanishes, the point A is called a punclum cofijugatum, 1 0. A point in which two branches of a curve terminate, is called a cuspia, 1 1 . The involute of any curve, is another curve traced by the end of a string in being unwound from the first curve, which is called the evolute. Thus, let A B C (fig. 1 46) be a curve having a thread laying in contact with every part of the same ; then, if the thread be un- wound from A, the curve line A D £« which its end will describe, will be the involute to the curve ABC, which latter will be the evo- lute of the <?tirve A D £. The principal lines connected with curves, such as tangenty ordinate^ absciBSoe^ &c., having been already defined in the first section of Chap. VI., it is unnecessary to recapitulate them here. 12. The equations of the conic sections have been already given, for the ellipse at page 165, the hyperbola at page 169, and the para- bola at page 171 : the equation for the circle is y = s/dx — aj*, d being the diameter ; the whole of these four curves are therefore lines of the second order, the quantities a; and y, only being raised to that power. We shall now pass on to describe the principal pro- perties of those curves which are most frequently employed for useful purposes. Sect. II. The Conchoid. Conchoid^ or Conchiles^ (from the Latin concha^ a shell,) is the name given to a curve by its inventor, Nicomedes^ about 200 years before the Christian era. The conchoid is thus constructed: AP and BD (fig. 147) being two lines intersecting at right angles: from P draw a number of other lines PFDE, &c., on which make always DE = DF = AB or BC; so shall the curve line drawn through all the points E, £, E, be the first conchoid, or that of Nicomedes; and the curve drawn through all the other points, F, F, F, is called the second conchoid; though, in reality, they are both but parts of the same curve, having the same pole P, and four infinite legs, to which the line DBD (called the directrix) is a common asymptote. The inventor, Nicomedes, contrived an instrument for describing his conchoid by a mechanical motion, in the following manner: in the ruler DD (fig. 148) is a channel or groove cut, so that a smooth nail firmly fixed in the movable ruler CA, in the point D, may slide freely within it: into the ruler AP is fixed another nail at P, for the movable ruler AP to slide upon. If, therefore, the ruler A P be so moved as that the nail D passes along the groove D D, the style, or point in A, will describe the first conchoid. CHAP. VII.] 0URVB8 : CI880ID. 1 77 CoDchoids of all possible Turieties may also be constructed with great facility by Mr. Jopling's apparatus for curves, now well known. 1. Let AB = BC == DE = DF = a, PB = ft, BO = EH = x, and OE = BH = y : then the equation to the first conchoid will be «* (ft -I- «)* + xy = a- (ft + «)*, or, a?* + 2ftd^ + ft«a?' + x'f =^0^1^ + 2a'fta? + a^«*; and, changing only the sign of x^ as being negative in the other curve, the equation to the second conchoid will be a?' (ft - xY -h a?V = a^ (* - x)\ or, «* — Zhx" + h^x^ + x^'f = a" ft- — 2a-ftx + aV. 2. Of the conchoids expressed by these two cauations (or rather the same equation with different signs), there are three species ; first, when a is lets than ft, the conchoid will be of the form shown in fig. 148, the inferior branch having a punctum conjugatum at A ; secondly, when a is equal to ft, the conchoid will assume the form shown in fig. 1 49, the punctum conjugatum giving place to a cuspU at B ; and thirdly, when a is greater than ft, the cuspis becomes a noduSj the conchoid taking the form shown in fig. 150. Newton approved of the use of the conchoid for trisecting angles, or finding two mean proportionals, or for constructing other solid problems. But the principal modem use of this curve, and of the apparatus by which it is constructed, is to sketch the contour of the section that shall represent the diminution of columns in ar- chitecture. Sbct. III. The Ciuoid or Cyewid. The eisMoid is a curve invented by an ancient Greek geometer and engineer named Diodes^ for the purpose of finding two continued mean proportionals between two given lines. This curve admits of an easy mechanical construction ; and is described very beautifully by means of Mr. Jopling's apparatus. At the extremity B of the diameter A B (fig. 151), of a given circle AOBo, erect the indefinite perpendicular eBE, and from the other extremity A draw any number of right lines, AC, AD, AE, &c., catting the circle in the points R, O, M, &c. ; then, if C L be taken = AR, DO = AO, EN = AM, &c., the curve passing through the points A, L, O, N, &c. will be the eissoid. 1. Here the circle AOBo is called the generating circle; and AB is the axis of the curves ALON, &c., A/ow, &c., which form a cuspii at A, and, passing through the middle points O, o, of the two semicircles, tend continually towards the directrix eBE, which is their common asjrmptote. 2. Letting fall perpendiculars LP, RQ, from any corresponding points L, R ; then is A P = BQ, and A L = CR. 178 curves: cycloid. [pabt i. 3. Also, AP : PB :: PL^ : AP*. So that, if the diameter AB of the circle = a, the abscissa A P = «, and the ordinate P L = y ; then « : a — jc :: y- : jr, or ar* = (a — *)y% which is the equation to the curve. 4. The arc AM of the circle = arc BR, and arc Ant = Br. 5. The whole infinitely long cissoidal space, contained between the asjrraptote ^ B E and the curves NO LA, &c., and A Lou, &c., is equal to three times the area of the generating circle AOBo. Sect. IV. The Cycioidy and Epicycloid, The cycloid^ or trochoid^ is an elegant mechanical carre first noticed by Descartes^ and an account of it was published by Jffr- senne, in 1615. It is, in fact, the curve described by a nail in the rim of a carriage-wheel while it makes one revolution on a flat horizontal plane. 1. Thu8,if a circle EPF (fig. 152),keeping always in the same plane, be made to roll along the right line A B, until a fixed point P, in its circumference, which at first touched the line at A, touches it again at B after a complete revolution ; the curve APVPB described by the motion of the point P is called a cychid. 2. The circle EPF is called the yeneraiing circle ; and the ri^t line A B, on which it revolves, is called the base of the cycloid. Also, the right line, or diameter CV, of the circle, which bisects the base A B at right angles, is the axis of the cycloid ; and the point V, where it meets the curve, is the vertex of the cycloid. 3 If P rfig. 153) be a point in the fixed diameter AF produced, and the circle A E F be made to roll along the line A B as before, so that the point A, which first touched it at one extremity, shall touch it again at B, the curve PVP, described by the point P, is called a curtate cycloid. 4. And, if the point P (fig. 154) be anywhere in the unproduced diameter A P, and the circle A EF be made to roll along AB from A to B; the curve PVP, is, in that case, called the inflected or prdaie cycloid. The following are the chief properties of the common cycloid. 5. The circular arc VE (fig. 155) is equal to the line EG between the circle and cycloid, parallel to A B. 6. The semi-circumference VEC is equal to the semi-base CB. 7. The arc VG equals twice the corresponding choinl VE. 8. The semi-cycloidal arc VGB equals twice the diameter VC. 9. The tangent TG is parallel to the chord VE. 10. The radius of curvature at V equals 2 C V. 11. The area of the cycloid AVBGA is triple that of the circle CEV; and consequently that circle and the spaces VECBG, V E' C A Q\ are equal to one another. CHAP. VII.] curves: QUADRATRIX. 179 12. A body falls through any arc LK (fig. 156) of a cycloid re- Teraed, in the same time whether that arc be great or small ; that is, from any point L, to the lowest point K, which is the vertex re- versed: and that time is to the time of falling perpendicularly through the axis M K, as the semi-circumference of a circle is to its diameter, eras 3-141593 : 2. And hence it follows that if a pendulum be made to vibrate in the arc L K N of a cycloid, all the vibrations will be performed in the same time. 1 3. The eoolvte of a cycloid is another equal cycloid, so that if two equal semi-cycloids OP, OQ (fig. 156), be joined at O, so that O M is equal to M K the diameter of the generating circle, and the •tring of a pendulum hung up at O, having its length equal to O K or the curve OP; then, by plying the string round the curve OP, to whieh it is equal, if the ball be let go, it will describe, and vibrate in the other cydoid PKQ; where OP equals QK and OQ equals PK. 14. The cycloid is the curve of swiftest descent : or a heavy body will fall from one given point to another, by the way of the arc of % cycloid passing through those two points, in a less time than by ma^ Olher route. Hence, this curve is at once interesting to men of sciattat aod to practical mechanics. 15. If iIm generating circle, instead of rolling along a straight line is made to ndl upon the circumference of another circle, the curve described by aiij point in its circumference is called an epicycloid, 16. This curve teives.its importance in practical mechanics, from its being the curve m«i| frequently adopted for the teeth of wheels. 17. If the generaduf circle revolves upon the convex circnm- ferenee of the quiescent cMe, as in fig. 157, it is then called an eMeriar epie^fdoid A B CD. B«t if it revolves on the concave circum- ference, as in ^g, 158, the curve h called an interior epicycloid. 18. The area of an epicycloid, either interior or exterior, may be found from the following proportion: as AB:3AB-hBC ::a6c: abD i: the area of the generating circle : the area of the epicycloid J>BEebD, 19. If the diameter of the generating circle is equal to half that of the quiescent circle, the epicycloid becomes a straight line; which circmnstance has been taken advantage of for converting a continued circular motion into an alternating rectilinear motion, in the manner explained in the article on ^* Select Mechanical Expedients," in the Appendix* Sbct. V. The Quadratrix. The quadrcUrix is a species of curve by means of which the quad- rature of the circle and other curves is determined mechanically. For the quadrature of the circle, curves of this class were invented by Dinoitrates and TechimltauseUy and for that of the hyperbola by Mr. Perks. We shall simply describe in this place the quadratrix of N 2 180 curves: catenary. [part i. Dinottraies; in order to show its use in the division of an arc or angle. To construct this quadratrix, divide the quadrantal arc AB (fig. 159) into any number of equal parts, A&, hc^ cd^ dB; and the radius AC into the same number of equal parts, A/^ /y, gh^ AC. Draw radii C^ Cc, &c., to the points of division upon the arc; and let lines /li pm, &c.y drawn perpendicularly to AC from the several points of division upon it, meet the radii in /, m, n, &c., respectively. The curve A/mnD that passes through the points of intersection I, fii,&c. is the quadratrix of Dinostrates. The figure A CD n ml A thus constructed may be cut out from a thin plate of brass, horn, or pasteboard, and employed in the division of a circular arc. Thus, suppose the arc IL (fig. 160) or the angle IKListobe divided into five equal parts. Apply the side AB of the quadra- trix upon I K, the point B corresponding with the angle K. Draw a line along the curve AS, cutting RL in F. Remove the instru- ment, and from F let fall the perpendicular FE upon IK. Divide EI into five equal parts, by Prob. VII. page 113, and through the points of division draw CM, DN, &c. parallel to £F. Then, through their intersections M, N, O, P, draw the lines KM, KN, KO, KP, radiating from K, and they will divide the angle IKL into five equal parts, as required. Note 1 . — If, instead of dividing the arc into equal parts, it were proposed to divide it into a certain number of parts having given ratios to each other; it would only be necessary to divide EI into parts having the given ratio, and proceed in other respects as above. Note 2. — If the arc or angle to be divided exceed 90 degrees, bisect it, divide that bisected arc or angle into the proposed number of parts, and take two of them for one of the required divittona of the whole arc. Sect. VI. The Catenary. The catenary is a mechanical curve, being that which is assumed ^fl by a chain or cord of uniform substance and texture, when it is hung"^ upon two points or pins of suspension (whether those points be-^^ in a horizontal plane or not), and left to adjust itself in equilibrio— * in a vertical plane. The catenarian curve was for a long while regarded as the proper form for the chains of a suspension bridge, but a slight consideration, of the above definition will show its inapplicability. For, since the^ chains of a suspension bridge ought never to be of uniform substance^ and have also in addition to their own weight to support that of the roadway, which latter is very differently distributed to the former; the form of the curve which the chains will assume is materially modified, and approaches more nearly to that of a parabola*. • See page 207* CHAP. VII.] CUBVBS: CATENARY. 181 Let AB (fig. 161) be the points of suspension of such a cord, AaC^B the cord itself when hanging at rest in a vertical position. Then the two equal and symmetrical portions AaCy CbB, both ex- posed to the force of gravity upon every particle, balance each other predaely at C. And, if one half, as C&B, were taken away, the other half, AaC, would immediately adjust itself in the vertical position under the point A were it not prevented. Suppose it to be prevented by a force acting horizontally at C, and equal to the weight of a portion of the cord or chain equal in length to CM; then is C M the measure of the tension at the vertex of the curve; it is also regarded as the parameter of the catenary. Whether the portion AaC hang from A, or a shorter portion, as aC, hang from a, the tension at C is evidently the same : for, in the latter case, the resist- ance of the pin at a, accomplishes the same as the tension of the line at a when the whole AaC hangs from A*. Let the line C M which measures the tension at the vertex = />, let Cd (or the deflexion as it is termed) ^=Xy ad (or the semi-span) = y, Ca=4r, CD = A, AB^d, CaA = C&B = i/. Then, l.^=^(hyp.l«g'-±-*-±^!£l±^) = ^(hyp. log ^±^) = ^(hyp. log t±£) te/>M.logten(45*» -f \%); where M == 2*30255851, the number by which the common loe- ■rithnis must be multiplied to obtain the hyperbolic logarithms f; and 8 = the angle which the tangent to the curve at the point a makes with the horizon. 2. If the angle 8 of suspension made between the tangent to the curve at A or B, and the horizon be 45°; then d : I :: 1 : 11346. 3. When /= 2rf, then h = 7966 d, and 8 = TT** 3'. 4. When the angle S of suspension is 56"^ 2B\ then />, a^ y, and 4r, are as 1, 0*81, 1-1995, and 1*5089 respectively. In this case f, the tension at the point of suspension, is a minimum with re- elect to jr. 5. Generally, y = — tan SM . log tan (i 8). Or, logy = log tan 8 -f log {log cot (i 8) — 10} + '^22157 - 10. This last formula serving to compute an approximative result. • ThiP may easily be determined experimenully, by letting the cofd hung very freely over a pulley at C, and lengthening or shortening the portion there suspended, until it keqM AaC, in its due position ; then is the portion so hang- ing beyond the pulley eqnnl in length to C M. t ^ce page 93. 182 curves: catrnaby. [pabt l 6. The distance of the centre of gravity of the whole cane 9#, from the vertex = J (a? + ^-^ — />)• 7. The radius of curvature — ^= — : this at the ^eitoxii rad. curv. = p, 8. When S and p are given ; then jr = j9 . tan 8 t s j9 . sec 8 « = p . sec (8 - I) = ^^ , — cos s y = J9M . log tan (45° + i 8), 9. When 8 and z^ or ^ /^ are given s then /? = ;3r . cot 8 ^ = JBT . cosec 8 X ^ z . oosec 8 . versin 8 y = M4f . cot8 . log tan (45° + ^8). 10. When 8 and y are given : then y ^ M . log tan (45° -f i 8) M . cos 8 . log tan (45° + i 8) y . tanS ^ '^ M . log tan (45° -f ^8) y . versin 8 *^ l( . cos 8 . log tan (45° + i s) 11. When X and y are given ; then log tan (45° -f j 8) _ y ^ sec 8 . versin 8 VLx* ^rom which 8 may he found hy an approximative process ^ alac^ X _ ^ _ « . sin 8 sec 8 — 1 ' " versin 8 ' versin 8 in all these ca^es t is determined in kn^ of chain or cord of wh^ the catenary is actually constructed. 12. To draw the caUnary mechanicaUy, — If the distance 10 (fig. 161) hetween the points of suspension, and the depth DCof t^ lowermost point, he given, hang one extremity of a fine imifa''^ chain or cord at one of the points A, and (letting the chain or e^ adjust itself as a festoon in a vertical plane) lengthen or shorten^ as it is held near the other end, over a pin at B, until^ when at r^ CHAP. VII.] CUBVBfi: CATBHAHr. 183 it just reacbes the point C : so shall the cord form the catenary ; and a pencil passed along the cord, from A by a, C, &, to B, will mark the curve upon a vertical board brought into contact with it. 13. All catenaries that make equal angles with their ordinates at their points of suspension are similar, and have j; to y a constant ratio : and of any two which do not make equal angles, but have X to y in different ratios, a portion may be cut from one curve similar to the other. Thus, let ACB and A'C'B' (fig. 162), be the two curves, of which A' C" B' is the flatter. Suppose them placed upon one axis D C C\ and the tangent T' S', to the lower curve, at B', the point of suspension, to be drawn. Then, parallel to T'S' draw another line TS to touch the other curve in b. Through b draw ba parallel to B^A^. So shall the portion aCb of the upper catenary be similar to the lower catenary A'C'B\ 14. With reference to the practical uses of the catenary, we may now blend the geometrical and the mechanical consideration of its properties. Taking any portion Cb (fig. 163) of the catenary, from the lowest point C; its weight may be regarded as supported by tensions acting in the tangential directions C N, by. The strains at C and b may be conceived as acting at the point of intersection N ; above which, therefore, in the vertical direction N O, the weight of the portion Cb may be conceived to act at its centre of gravity, G. Hence, strain at C : weight of C^ :: sin ON6 : sin &NC :: cos&NR : sin 6 N R :: radius : tan &NR :: radius : tan </^ N. Hence, the horizontal tension at C being constant^ the weight, and consequently the length of any portion ci of the uniform chain must be proportional to the tangent of the inclination of the catenary to the horizon at the extremity b of the said portion. This may he ttgarded as the characteristic property of the catenary, 15. In like manner, the horizontal strain at C : oblique strain at & :: sin ^ N O : sin C N O :: cos & N R : radius :: radius : sec & N R. Therefore, the strain exerted tangentially, at any point 6, is pro- portional to the secant of the inclination at that point. Also, from § 14 and § 15, tangential strain at b : weight of B & C :: wecdby : tani^N. These properties evidently accord with the preceding equations. 16. Let, then, CO, in the axis produced downwards, be eqnal to the parameter, or the measure of tne horizontal strain at C ; and upon O as a centre with tadius CO dcscHbe a circle. A tangent dt drawn to this circle from cf, will be parallel to the tangent &NA of the curve at the point b to which db ib the ordinate. That tangent dt (to the circle} will also be equal in lencth to the corresponding por- tion 5 C of die curve : while the tension at b will be expressed by a length of the chain equal to the secant O d. So again, if D T be a tangent to the circle drawn from D, it will be equal in Icneth to B5c, and parallel to the tangent to the catenary at B; while the secant OD will measure the oblique tension at B; evidently, ex- eeeding the constant horizontal tension or strain at C, by the abadna CB. 184 CUBVBS: CATENAHY. [pART I. 17. When the parameter of the catenary, or the line which mea- sures the tension at the lowest point, is equal to the deflexion DC; if each of these he supposed equal to 1, then A B = 12*6339, the length of chain A C B = 3*4641 ; the strain at the points of suspen- sion A and B will each he 2, that at the lowest point heing 1 ; and the chain at A and B will make an angle of 60^ with the horizon. 18. If the strain at C he equal to the weight of the chain, and each denoted by 1: then AB = *96242, DC = -1180340, the tension at A or B = 1*118, the angle of suspension at those points nearly 26° 34'; the width of the curve is 8*1536 times, and the length 8*4719 times, the deflexion DC. 19. If the strain or tension at the lowest point be doable the weight of the chain : then if the parameter be 1, A B will be '49493, C D = 03078, the strain at A or B 1*03078, the angle of suspenaon about 14° 2', the width or span 16*0816 times, and the length of chain 16*2462 times the deflexion. The magnitudes of the lines, angles, and strains in many other cases, may be seen in the table below. The whole theory may be yerifled experimentally, by means of spring steelyards applied to a chain of given length and weight, placed in various positions, accord^ ing to the method suggested at page 244, when trc»Etting of the me- chanical powers. 20. Taking A B = </, C D = A, length A C B = /!» strain at C or parameter = p, then, in all cases where the deflexion is small com- pared with the length of the chain. Professor Leslie shows*, that d' d- p = - , -f ] A .... strain atAorB = --r-flA or /? = — — I ^ .... strain atAorB=-7 + JA ^ = ^■^33- In this case, too, the strains at G and A or B are nearly in the inverse ratio of the deflexion t. 21. The following table is abridged from a very extensive one given by Mr. WarCy in his " Tracts on Vaults and Bridges." Two examples will serve to illustrate its use. Ex. 1. Suppose that the span of a proposed suspension bridge is to be 560 feet, and the deflexion in the middle 25^ feet; what will be the length of the chain, the angle of suspension at the extemities, and the ratio of the horizontal pressure at the lowest point, and the oblique pressures at the points of suspension, with the entire weight of the chain ? * Elements of Natural Philosophy, p. 63. f For a very complete investigation uf the proper forms of catenaries for tns- pension bridges, with remarks on the Menai Bridge, and on the failure of the suspension bridge at Broughton, see Mr. Eaton Hodgkinaon*s paper in the Memoirs o( the Manchester Society, vol. v., New Series. CHAP. VII.] CURVXS: CATBNARY. 18^ TaUe ofBdaiions of Catenarian Curves^ the Parameter being de^ notedly 1. wiSS^ Dtfloioii. or DC. 8«»^.or Laofth ot chain, or CbD. TAMkmat the point at Semi-ipan di. Tided "by the deflexion, or DB fiC. !• C •00015 •01745 -01745 1-0001 114-586 2 •00061 •08491 -03492 10006 67-279 8 •00187 •05288 •05241 1-0014 88-171 4 •00244 •06987 •06993 10024 28-613 5 •00882 •08788 •08749 1-0088 22-874 6 •00551 •10491 •10510 1'0055 19046 7 •00751 •12248 •12278 1'0075 16-309 8 •00988 •14008 •14054 1*0098 14'254 9 •01247 •15778 -15838 10125 12-654 10 '01548 •17542 -17683 1'0154 11-872 11 •01872 -19818 •19438 10187 10-820 12 •02284 -21099 •21256 1-0223 9-444 18 •02680 •22887 •28087 10263 8-701 14 -03061 •24681 -24983 1-0806 8-062 15 •08528 •26484 •26795 1-0853 7-608 16 •04080 •28296 -28675 10408 7-021 17 -04569 •80116 •80573 • 10457 6-691 18 •05146 •81946 -32492 10515 6-208 19 •05762 •88786 -34433 1*0576 5-868 20 •06418 •85687 -36897 1'0642 5-563 21 •07114 •87502 •38386 1-0711 5-271 22 •07858 •89876 -40408 1-0786 5-014 28 •08686 •41267 -42447 1-0864 4-778 24 •09484 •48169 •44528 1^0946 4-562 26 •10888 •45087 •46681 11084 4-361 28 •11260 •47021 •48778 1'1126 4-176 28 •18257 •50940 •58171 1-1826 8-843 80 •15470 •54980 •57785 ia547 8-561 82 4 •18004 -59120 •62649 1-1800 8-284 84 16 •21003 •68710 •68180 1'2100 8-034 86 52 •24995 •69820 •74991 1-2499 2-773 89 11 •29011 •74480 •81510 1-2901 2-567 41 44 •84004 •80290 -89201 1-8400 2-362 44 •89016 •85660 •96569 1'3902 2-196 46 1 •48999 •90660 1*03610 1-4400 2-060 48 11 •49981 •96230 1-11780 1-4998 1-925 50 8 •56005 1-01420 1-19740 1-5800 1-811 52 9 •62978 1-07060 1-28690 1-6297 1-699 54 18 •71021 1-18040 1-88740 1-7102 1-592 56 28 •81021 119950 1-50890 1-8102 1-481 58 8 •88972 1-25100 1-60340 1-8897 1-416 60 1^00000 1-81690 1-78210 2-0000 1-817 64 6 1-28940 1-47020 2-05940 2-2894 1-140 67 28 1-60950 1-61350 2-41020 2-6095 1-002 67 82 161680 1-61640 1 2-41820 2-6168 0-9998 186 curves: catenary. [part i. Here D B — D C = -280 -h "25 875 = 10'8*2, a namber which is to be foand in the table. Opposite to that namber^ we find 11° for the angle of saspension, DB= 19:318, CB= 19488, tension at A or B = 10187, the constant tension at the Tertex being 1. (Fig. 161). CoDseqnentlj, 19318 : 19438 :: 560 : 563*48 length of the chain. Also, horizon, pressure at C being taken as 1*0000 the oblique pressure at A or B will = 1*0187 and the entire weight of chain will = '39876 Ex. 2. Sappoae that while the span remains 560, the deflexion is increased to 51. Here D B -f- D C = "284) -;- 51 = 5-49. This number is not to he found exactly in the table. The nearest is 5*553 in the last column, agreeing with *20^, the angle of suspension. Now, 5*55 — 5*49 = 06, and 5*55 — 5'27 = '28, the former differenee being oearlj one«>fifUi of the latter. Hence, adding to each number, in the line agreeing \iith '20°, one-fifth of the differ- ence between that and the corresponding number in the next line, we shall have An^e of suspension = aO"" 1*2', DC = 06556, DB = -86010, CD = -36797, tension at A = 1*10656. Hence -36010 : 36797 :: 560 : 572 24, length of chain. Also, horixontal pressure at C being taken as 1*0000 the oblique prest^ure at A or B will ss 1*10656 and the entire weight of chain will = -73594 Comparing this Tilth the former case, it n-ill be seen that the tensions at C and A, in reference to the weight of the chain, arc diminished nearly in the inrerse ratio of the two values of DC; thus confirming the remark in art. '20. In making use of this table, the remark at page 1 80 must be borne in mind, since the results obtained from it, will be only approxima- tions more or less correct, as the weight of the chain exceeds that of the roadway and load. Where greater accuracy is required, the formula given at page 208 must be employed in its stead. PT. II. CH. I.] mechanics: 8TATIC8. 18? PART II. MIXED MATHEMATICS. CHAP. I. MIOHANIOS IN OBNBRAL. 1. Meehanies is ^e science of equilibrium and of motion. 2. Every cause which tends to move a body, or to stop it when in motion, or to change the direction of its motion, is callea a force or 3. The dkreetion of a foree is that straight line in which the point to which the force is applied tends to move by virtue of that force. 4. When the forces that act upon a body, destroy or annihilate each other s operation, so that the body remains quiescent, they are «id to be in tquiUhrium^ and are then called presntres, 5. The dirmticn of a pressure is the straight line in which that presaore tends to prevent the motion of the point to which it is ap- pKad. 6. The effect of anv force or pressure is found to be the same, at wfaattfver point in the line of its wection it be applied. 7. Two forces or pressures are said to be tqualy when, being ap- plied in appoiite directions to the same point, no motion ensues. If, however, both forces are supposed to act in the iame direction, the single force or pressure which would be required in the opposite direction to keep the point in equilibrium, is said to be double either of the former forces. And one of the former forces being taken as the unit, the latter force would be represented by two of such units. 8. When two forces can be expressed in terms of the same unit, the^ are said to be eommensurcAle. 9. It IS usual to represent forces and pressures by lines, the diree- iioH of the line coinciding with the actual direction of the force, and (a line of a certain length being taken as the representative of a unit of force,) the ien^ of the line expressing the amount or magnitude of the given force or pressure ; and a force thus denoted is said to be represented both in magnitude and direction, 10. When several forces or pressures act in different directions upon the same point, it is possible to find the direction and magnitude of another force or pressure which would replace all the others ; that 188 mechanics: statics. [pabt ii. is, if the others were remored, and this one force applied in their stead, the effect produced would he the same. Sach a force is called the resultant of the others, which are named the components^ and the operation which we have supposed is called the composition of forces, and its converse, or finding the directions and magnitudes of any nnmher of forces which would produce the same effect as any one given force, is called the resolution of forces. 11. The moment of a force or pressure ahont any point, is the perpendicular distance from that point to the line of direction in which the force acts, multiplied hy the numher of units expressing that force. 12. Vis inertia^ or power of inactivity, is defined hy Newton to be a power implanted in all matter, by which it resists any change attempted to be made in its state, that is, hy which it requires force to alter its state, either of rest or motion. 13. Vis vioiiy or living force, a term used by Leibnitz to denote the force or power of a body in motion; or the force which would be required to bring it to a state of rest. 14. Mechanics is usually divided into five branches; viz. — I. Statics^ which relates to the equilibrium of pressures applied to solid bodies, and of the weight and pressure of bodies when at rest. II. Dynamics^ which relates to tlie motion prodaced in solid bodies by the application of force. III. ffydrostaticsy which relates to the equilibrium and pressure of non-elastic fluids, and of the weight, pressure, and sta- bili^ of bodies immersed in them. IV. NydrodynamicSy which relates to the motion of non'-elaaie fluids by the application of force. V. Pneumatics^ which relates to the various circumstances at- tending the equilibrium or motion of elastic fluids. CHAP. II.] STATICS. 189 CHAP. II. STATICS. Sect. I. Statical Equilibrium, 1. Ip any two pressures applied to a point keep it in eqailibrinro, they most be eqnal to each other, and must act in the same straight line and in contrary directions. 3. If any tkree pressures applied to a point keep it in equilibrium, they must all act in the same plane ; and any one of those pressures is represented in magnitude and direction by the diagonal of a paral- Ukaram^ whose sides represent the other two pressures in magnitude and direction. 3. If nnj/our pressures whose directions are not in the same plane, applied to a point, keep it in equilibrium, any one of those pressures is represented in magnitude and direction by the diagonal of a paral^ lelapipedon^ whose contiguous edges represent the other three pres- sures in magnitude and direction. 4. If any number of pressures whose directions are all in the same plane, applied to a point, keep it in equilibrium, those pressures will be represented in magnitude and direction by a polygon whose M&i are made parallel to the direction of those pressures, and pro- portional to their magnitude. 5. If any number of pressures in the same plane be in equili- brium, and any point be taken in that plane from which their mo- ments are measured, then the sum of the moments of those pressures which tend to turn the plane in one direction about that point, will be equal to the sum of the moments of those which tend to turn it in the opposite direction. 6. If any number of pressures acting in the same plane, and which are in equilibrium about a given point, be moved parallel to the direc- tions in which they act, until they all intersect in any given point, they will still be in equilibrium about that point. 7. The resultant of two parallel pressures, if acting in the same direction, is equal to their 8um; but, if acting in contrary directions, is equal to their difference^ the direction of their resultant being in the direction of the ereater pressure. And generally the resultant of any number of parallel pressures is equal to the several pressures added together with their proper signs. 190 STATICS. [part II. From the foregoing propositions we may deduce the following for- mulffi: — 8. Let P. and P^ represent as j two pressures, and ff the angle formed hy tneir two lines of direction ; kt R be their resultant ; y the angle which its line of direction makes with that of Pj, and ^ that which it makes with P^; then R = >/Pi* + Pj' + 2 Pj Pj cos /9 . . . (I.) P.. sin $ ,-- . tan y = - — ~- ^ (II.) '^ Pi + P« cosg ^ ^ Pj : Pg :: sin ^ : sin y (IH.) or, three forces being in equilibrium, any two of them are to each other inversely as the sines of their inclinations to the third. 9. To determine the resultant of any number of pressures in the same plane: — Let P,, P^, P,, &c. represent the pressures in magni- tude, and a J, a^, a 3, the inclinations which their lines of direction make with some line given in position ; let R be their resultant, and » its inclination to the same line; then R cos » = P| cos a, -f- Pg cos aj -f .... + P« COS «« . (IV.) R sin » = Pj sin a, + P^ sin a^ + . . . . + P« sin «, . . (V.) in which the several terms are taken positive or negative according to the direction in which they act. From these we obtain R = {>/(P, sinai+Pjsina, + + P,sin«,)* + (Pi cos a, + Pg cos «^ -f + P, cos aj«} . . . (VL) t^„,^P|Bina, 4-P.8ina,.f.... + P,«i°a, ...(VU.) P, cos a J 4- Pg COS«j 4- . . . . -h P^ COS «, 10. Let Pj, Pg, P., &c., be any number of parallel pressures, (being + or — according to their direction,) and D,, D^, D^, Ac, be their perpendicular distance from any given parallel plane : let B be their resultant, and A its perpendicular distance from the same plane: then R = P, + P2 -f + P (VIII.) ^^P»I>>-i-P,D,+ 4-P,D, Pi + P« + . . . -h P. ^ ' As these formulae and propositions are of universal application in the constructions of civil engineers, architects, and mechaniciaos, we shall give a few simple examples, to render tlieir use clear to those who are not familiar with mathematical form'ulee. Ex, 1. Suppose that a weight B is attached by a stirrup to the foot . of a king-post A B, which is attached to two rafters A C, A D, in the respective positions shown in fig. 1 64. Then if A E be set off upon CHAP. II.] STATICS. 191 AB, equal in numerical value to the yertical strain upon A B, and the parallelogram AFEO be completed, AF measured upon the same scale will show the strain upon the rafter A C, and A O the strain upon A D. Ex. 2. Let it be proposed to compare the strains upon the tie- beams AD, and the struts AC, when they sustain equal weights B, in the two different positions indicated by figs. 165 and 166. Let A £ in one figure be equal to the corresponding yertical line A£ in the other, and in each represent the numerical value of the weight B, that hangs from A. Through E in both figures, draw lines parallel to D A, A C, respectively, and let them meet A C, and D A produced in F and O: then AFEG in each figure is the parallelogram qf/orces by which the several strains are to be measured. A G represents the tension upon the tie-beam A D, and A F the strain upon the strut A C. Both these lines are evidently shorter in the lower figure than they are in the upper, A E being of the same length in both : there- fore the first figure exhibits the least advantageous position of the beams. It is evident also, that while C A tends upwards and D A downwards, the greater the angle D A C, the less is its supplement C A G, and the less the sides F A, A G, of the parallelogram. JSm. 3. Let it be required to determine geometrically, and by com- potadon, the magnitude and direction of the four pressures P^, P^, Pj and P^, (fie. 167,) all applied to the point A, and acting in the sme plane; let Pj = 12, P^ = 9, P, = 16, P^ = 15, and the angles which their directions make with the line BAG, (given in pontion,) be 77^ 37% 9° and 48% respectively. 1. CrecmeiricaUy. — From the point A (fig. 168) lay off the angle CAD equal 77% and upon the line A D set off 12, from any scale of equal parts ; then through D draw the line D E, equal to 9 parts by the same scale, and making an angle of 37° with B A C ; again, through £ draw the line £ F equal to 16, and inclined 9° to B A C ; and lastly, through F draw £ G, equal to 15, and inclined 48° to B A C; then the straight line AG, connecting the first and last points, and completing the polygon A D £ F G A, will be the resultant of the four given pressures ; and being measured by the same scale, will be found equal to 36*72 parts nearly, and to be inclined 13° to B A C. 2. By Computation.— k^ the pressures P^, Pg) and P., are not on the same side of the line BAC (fig. 167) as P^, the signs of their ftfies most be taken differently ; but as they are ail on the same side of the line H A I, (perpendicular to B A C,) the signs of their cosines must be taken alike; then if R is their resultant, and » its inclination to B A C, we obtain from formula (VI.) — E=^/{(ia X -9744 + 9 X -6018+ 16 X 1664-15 x •7431)' -f (12 X -225 + 9 X -7986 -f 16 x -9877 + 16 x -6691)*} = 36-7138; •ad from formula (VII.) — 12 X -9744 + 9 X -6018 + 16 x 1564 - 15 x '7431 ^^,,,^ ttn«B3 . ='23106 12 X -225 + 9 X -7986 + 16 x -9877 + 13 x 6691 .-. • = 13^ In which the values are the same as were found geometrically. 192 CENTRE OF GRAVITY. [PABT II. Sect. II. Centre of Gravity, 1. Gravity is the force in virtue of which bodies left to themselves fall to the earth in directions perpendicular to its surface. 2. We may distinguish between the effect of gravity and that of weighty by observing that the former is the power of transmitting to every particle of matter a certain velocity which is absolutely inde- pendent of the number of material particles ; while the latter is the effort which must be exercised to prevent a given mass from obeying the law of gravity. Weighty therefore, depends upon the fiuu$; but gravity has no dependence at all upon it. 3. The centre of gravity of any body, or system of bodies, is that point about which the body or system, acted upon only by the force of gravity, will balance itself in all positions: or it is a point which, when supported, the body or system will be supported, however it may be situated in other respects. The centre of gravity of a body is not always within the body itself; thus the centre of gravity of a ring is not in the substance of the ring, but in the axis of its circumscribing cylinder; and the centre of gravity of a hollow staff, or of a bone, is not in the matter of which it is constituted, but somewhere in its imaginary axis; every body, however, has a centre of gravity, and so has every system of bodies. 4. Varying the position of the body will not cause any change in the relative position of the centre of gravity ; since any such muta- tion will be nothing more than changing the directions of the forces, without their ceasing to be parallel ; and although the amount of the forces may not continue the same, in consequence of the body bein? supposed at different distances from the earth, still the forces upon aO the molecules vary proportionally, and the position of their centre re- mains unchanged. 5. When a heavy body is suspended by any other point than its centre of gravity, it will not rest unless that centre is in the same vertical line with the point of suspension ; for in all other positions the force which is intended to ensure the equilibrium will not be di- rectly opposite to the resultant of the parallel forces of gravity upon the several particles of the body, and of course the equilibrium will not be obtained. (See Art. 9, on Pendulums^ page 222.) 6. If a heavy body be sustained by two or more forces, their directions must meet, either at the centre of gravity of that body, or in the vertical line which passes through it. 7. When a body stands upon a plane, if a vertical line passing through the centre of gravity fall within the base on which the body stands, it will not fall over; but if that vertical line passes without the ba^e, the body will fall, unless it be prevented by external means. When the vertical line falls upon the extremity of the base, the body may stand, but its state (which is called unstable equilihriuni) may be disturbed by a very trifling force ; and the nearer this line passes to any edge of the base the more ensily may the body be thrown CHAP. II.] CBNTRB OF GRAVITY. 193 OTer ; the nearer it falls to the middle of the hase, the more firmly the body stands. Upon this principle it is that leaning lowers have been bnilt at Pisa, and various other places ; the Tertical line of direction from thecentre of gravity falling within the base. And, from the same principle it may be seen , ^^^ that a waggon loaded with ^fl^|p^|^ heavy materials, as B, may stand with perfect safety, on the side of a convex road, the vertical line from the centre of gravity falling be- tween the wheels ; while a waggon A with a high load, as of hay, or of wool-packs, shall fall over, because the vertical line of direction fells without the wheels. 8. Owing to the great distance of the earth's centre from its sar- &ce, the directions of the force of gravity of the several molecule composing a body may be considered parallel without any appreciable error, and therefore all that has been said in the foregoing section on the subject of parallel pressures will apply to the force of gravity ; and the formula for finding the position of the resultant of any num- ber of pandlel pressures, may be applied to determine the position of the centre of gravity. Thas, if Bj, B^, B3, &c., denote the weights of the particles of any body, and D^, D„, D,, &c., the perpendicular distances of their reflective centres of gravity from any given plane ; then, the dis- tance TA) of their common centre of gravity from the same plane is found by formulsB (IX.) to be ^ ^ B^D, +B^ Dg -I- +B,D, Bj +B2 + +B. Therefore, if by means of this formula we ascertain the distance of the centre of gravity from any three planes given in position, we shall have determined its exact situation in space. 9. The common centre of gravity of two bodies divides the right line which joins their respective centres of gravity, in the inverse ratio of their weights. The centre of gravity of any number of bodies may be found, by finding the common centre of gravity of any two of the bodies, and then considering this as the centre of gravity of a body equal to the combined weight of the two, we may find the com- mon centre of gravity of this imaginary body and a third ; and thus proceed, ad libitum, 10. If the particles or bodies of any system be moving uniformly and rectilineally, with any velocities and directions whatever, the centre of gravity is either at rest, or moves uniformly in a right line. Hence, if a rotatory motion be given to a body and it be then left to move freely, the axis of rotation will pass through the centre of 194 CIKTRB OF ORAYITT. [PABT H. grwnty : for that centre, either remaining at rest or moving nniformly forward in a right line, has no rotation. Here too it may be remarked, that a forte appUed at the centre of ^ravit^ of a body^ eannct produce a rotaUny motion. 11. To find the centre of gravity meebanically, it is only requisite to dispose the body successiYely, in two positions of eqoilibriam, by the aid of two forces in vertical directions, applied in succession to two differMt points of the body; the point of intersectkm of these two directions will show the centre. This may be exemplified by particularising a few methods. If the body have plane sides, as a piece of board, hang it up by any point (A fig. 169), then a piuiAb-line suspended from the same peine will pass through the centre of gravity; therefore toark that line (AB) upon it : and after suspending the body by another point (C), apply the plummet to find another such line (C D) ; then will th^ interaeetion show the centre of gravity. Or thus : hang the body from a tack successively by two strings attached to different parts of it, and each time mark upon it the^line of a plummet attached to the same tack ; then will the intersection of the two lines be the centre of gravity. Another method : Lay the body on the edge of a triangular prism, or such like, moving it to and fro till the parts on both sides are in equilibrio, and mark a line upon it close by the edge of the prism: balance it again in another position, and mark the fresh line by the edge of the prism; the vertical line passing through the mterseetion of these lines, vrill likewise pass through ^e centre of gravity. The same thing may be effected by laying the body on a table, till it ie just ready to fall off, and then marking a hue upon it by the edge of the table : this done in two positions of the boay will in like manner point out the position of the centre of gravity. Wlien it is proposed to find the centre of gravity of the arch of a bridge, or any other structure, let it be laid down accurately to aoale upon pasteboard ; and the figure being carefully cut out, its centre of gravity may be ascertained by the preceding process. 12. The centre of gravity of a right line, or of a paraUdlogram, prism, or cylinder, is in its middle point; as is also that of a circle, or of its circumference, or of a sphere^ or of a regokr polygon ; the centre of gravity of an ellipse, a paraboks a cone, a conoid, a sj^ roid, &c., is situated in its axis. And the same of all symmetrioal figures. 13. The centre of gmvity of a triangle is the point of intersectioa of lines drawn from the three aneles to the middles of the sides re* spectively opposite: it divides ea(m of those lines into two portions ia tlie ratio of 2 to 1. 14. In a Trapemum. Divide the figure into two triangles by the diagonal AC (fig. 170) and find the centres of grarity £ and F of these triangles; join £ F, and find the common centre O of these twe by this proportion, ABC : ADC :: FO : EG, or ABCB : ABC :: £F : £0. Or, divide the figure into two triangles by a dii^nal CHAP. II.] CmRB OP GRAVITY. 195 BD; then find thm centres of gravity; and the line which joins them will intersect £F in O, the centre of grayity of the trapezium. 15. In like manner, for any other plane fignre, whatever he the numher of sides, divide it into several triangles, and find the centre of gravity of each ; then connect two centres together, and find their common centre as ahove; then connect this and the centre of a third, and find the common centre of these; and so on, always connecting the last found common centre to another centre, till the whole are included in this process; so shall the last common centre be that which is required. 16. The centre of gravity of a circular arc is distant from the centre of the arc a fourth proportional to the arc, the radius, and the chord of the arc. 17. In a circular sector, the distance of the centre of gravity from 2 c r the centre of the circle is— — ; where a denotes the arc, e its chord, oa and r the radius. 18. The eentres of gravity of the mrfaees of a cylinder, of a cone, and of a conic frustum, are respectively at the same distances from the origin as are the centres of mvity, of the parallelogram, triangle, and trapezoid, which are verticsl sections of the respective solids. 19. In the segment of a sphere, or spheroid, A being the whole ans, and k the height of the segment, the distance of the centre of A A ._ dk h gravity from the vertex is equal -^--^ 7-7 ; and when the height be- o A — 4 A oomes half A, or the segment becomes a hemi-sphore, the distance Irom the vertex is equal to ^ A. 20. The centre of grarit^ of the surface of a spheric segment is at the middle of its versed sine or height. 21. The centre of gravity of the convex turface of a spherical lone, ia in the middle of that portion of the axis of the sphere which is intercepted by the two bases of the zone. 22. In a cone, as well as any other p3rramid, the distance from the vertex is \ of the axis. 23. In a conic frnstnm, the distance on the axis from the centre of the leaser end, is J A. -j- : whore h equals tlie he^fat, and R r the radii of the greater and lesser ends. 24. The same theorem will serve for the frustum of any regular pyramid, talcing R and r for the sides of the two ends. 25. In the paraboloid, the distance from the vertex is f of the axis. 26. In the frustum of the paraboloid, the distance on the axis from 2 R* + r* the centre of the lesser end, is J h. - : where h equals the hcigfat, and R r the radii of the greater and lesser ends. 2 196 IQUILIBBIUM OF PIBB8. [PABT H. 27. Every figure, whether superficial or solid, generated bj the motion of a line or surface, is equal to the product of the generadng magnitude multiplied bj the path of its centre of gravity. As an example, let A B C (fig. 171) be a right angled triangle, the revolution of which on the leg A B will produce a cone ; let A B =: 9, BC = 6, and D be the centre of mvity; then by § 13, AD ^ | AF, therefore D£ sfFB = Q equal the radius of the circular path de- scribed by the centre of gravity, the circumference of which will therefore be 4 x 31416 = 125664, which multiplied by (3 x 9) = 27, the area of the triangle, will be 339*2928, the solid content of the cone; and by the rule given at page 130 for finding the solidity of a cone, we obtain 12* x -7854 x 3 = 3392928, the same result. Sect. III. General application of the principles of StaUcs to the equilibrium of Structures, Every structure is exposed to the operation of a system of forces; so that the examination of its stability involves the application of the general conditions of equilibrium. Now, no part of a structure can be dislocated, except it be either by a progressive, or a rotatory motion. For either the part is dis- placed, without changing its form^ in which case it is (as a system of invariable form) incapable of receiving any instantaneous motion, which is not either progressive or rotatory ; or else it happens to be displaced, changing at Vie same time its form^ which, considering the cohesion of its parts, cannot take place without its breaking in its weakest section ; in which case a progressive motion is generated if the force acts perpendicularly to the section, and a rotatory motion, if it acts obliquely. We shall here consider the most useful cases ; indicating by the word stress^ that force which tends to give motion to the structure, and by resistance^ that which tends to hinder it. EQUILIBRIUM OF PIERS, OR ABUTMENTS. 1. If we suppose figure 1 72 to be the vertical section of a pier, we may reason upon that section instead of the pier itself, if it be of uniform structure. Let O be the place of the centre of gravity of the section A B C D, S Z the direction in which the stress acts, meeting X I, the verticai line through the centre of gravity, in I. Then, supposing the stress to be resolved into two forces, one (Pj) vertical, the other (P^) horizontal ; the pier (regarding it as one bodv) can only give way either by a progressive motion from B towards A, or by a rotatory motion about A. 2. The progressive motion is resisted by friction. If W denote the weight of the pier, Pj the stress estimated vertically, and Pj its horizontal effort, then the pressure on the base sW + P^, and CHAP. II.] EQUILIBRIUM OF PIERS. 197 potting /for the coefficient of friction, its friction =/(W + Pj, which 18 the amount of the resistance to progressive motion. So that to ensure stahilitj in this respect we must have /(W + PJ> P, (I.) 8. To ensure stahilitj in regard to rotation, taking the moments of the forces ahout the point A, we must have W . AX -f Pj . Xe > P, . Tq (II.) Or, supposbg the stress not to he resolved, its moment ahout tho point A must be less than that of the weieht of the pier about the same point ; that is, putting S for the whole stress, W.AX>S.aY (III.) Or, by a graphical process, suppose the two forces W and 8 to he applied at I, and complete the parallelogram, having sides which represent these forces. Then must the diagonal representing the resultant of those forces produced meet the base on the side of A, towards B, to ensure stability. 4. If, as is very frequently the case, the vertical section of the pier is a rectangle, putting k for the height of the pier, h for its base, w for the weight of a cube unit of the material of which it is formed, and R a horizontal stress applied at the summit of the pier ; then the pier will be in a state of unstable equilibrium, as far as regards its progressive motion, when hfhw^lBi :.... (IV.) and as r^grds its being overthrown when V=* (V) But in order that the structure may really be secure, these equations most become b/hw>B^ and — ->R, and the more the first member exceeds the second the greater will be the stability of the pier. EwampU 1. Suppose a rectangular wall 89*4 feet high, and of a material weighing 125 lbs. per cube foot, is to sustain a horizontal atnin of 99(K) lbs. avoirdupois on each foot in length, applied at its somiDit: what must be its thickness that there may be an equilibrium, taking /= 1*75. By transposing formula (IV.) above, we obtain and substituting the several values given above, we have h = = 2-68 feet; •75 X 89-4 X 125 198 PRESSURI OF EARTH AOAlVWt WALLS. [PABT II. therefore the thickness of the pier at its base mast be more than 2*68 feet, otherwise it will be moved horizontally by the stress R. But we must also inquire, what thickness is requisite to prevent the pier being overthrown; and this we shall ascertain from formula (v.), which by transposition becomes V2R /2 X9900 __^ and * = A / — r^T — = 12*58 feet. Therefore, we see, that the pier most be more than 12*58 feet upon its base, otherwise it will be overthrown by the horizontal strain at R. Here, as the thickness required to prevent overturning is much the greatest, the computation m reference to the other kind of equili- brium may usually be avoided. Example 2. An embankment, or dam, A B D (fig. 17d), formed of clay between two rubble walls, is exposed to the pressure of 80 feet of water ; it is required to ascertain whether the dam will be overthrown, and also whether it will slip upon its bed. Its dimen- sions are, D C = 4 feet, B C which is vertical = 32 feet, and A B Bs 12 feet; its weight may be taken at 135 lbs. per cubic foot, and the coefficient of friction or f ^ '5. First, by the method laid down in § 14, page 194, we ascertain the distance of the centre of gravity of the wall from its interior face B C, to be 4*333 ; therefore the distance (A X) of its line of direc- tion from A will be 7*667 feet. The weight of the wall (W) will be 12+4 — - — X 32 X 135 = 34560 lbs.; the pressure of the water 30 (Pg or 8) will be -— X 62*5 = 937*5, and the centre of pressure of the water being at two-thirds of its depth (as explained at page 250), the distance (AZ) of its line of direction above the point A will be 10 feet. Now, in order to ascertain its stability to resist progressive motion, we must substitute these values in formula (I.), when we obtain •5 X 34560 > 937*5, or 17280 > 937-5; therefore there would be no fear of the embankment slipping upon its base. And from formula (III.) we have, 34560 X 7-667 > 937-5 X 10, or 264972 > 9375 ; therefore the wall cannot be overthrown. PRE88URB OF EARTH AGAINST WALLS. 1. Let DACB ^fig. 174) be the vertical section of a wall behind which is posited a oimk or terrace of earth, of which a prism whose CHAP, n.] raammR of xabth against walls. 199 section is represented by CBH wonid detach itself and fall down, were it not preyented by the wall. Then B H is denominated the line of rupture or the natural slope^ or natural deeltvity. In saudy or loose earth, the angle CHB seldom exceeds 30°; in stronger earth it becomes 37°; and in some favourable cases more than 45°. 2. Now, the prism whose vertioal section is C B H, has a tendency to descend along the inclined plane B H by reason of the force of gravity ; bat it is retained in its place, not only by the force opposed to it fy the wall, bat also by its cohesive attachment to the face B H, aDd by its fiicdon upon the same surface. If we resolve the weight of this prism into two forces, one acting pcfpendicular to the plane B H, and the other parallel to the same, the latter minus the force required to overcome either the cohesion or the friction of the surface B H, will be the strain acting in a direc- tion passing through the centre of gravity of the prism CBH and parallel to B H, upon the back of the wall, and tending to overthrow It ; and the aiBount of this atraio, and its direction, being ascertained, the formulsB already given may be employed to determine the sta- InU^of the wall, I H 2s LH + eohesioii + friction. S. It is evident, therefore, that m the angle at whidi the earth will stand U one <>f the elements in the calculation, the solution to this inquiry must be, in a great measure, experimental. It has been found, however, theoretically, bv M. Prony*, and confirmed experi- mentiJly, that the angle formed with the vertical (CBH) by the prism ojf earth that exerts the greatest horizontal stress against a wall, is half the angle which the natural slope of the earth makes with the vertical : and this emrious result greatly amplifies the whole inquiry. Puttisg A for the heiffht of the wall, ff for the angle CBF, or half C B H, the natural slope of the ground, w^ for the weight of a cabit onit of the ground, and /for the coefficient of friction we have h^w^ . tan/S_ 2 the weight of the prism CBF; and resolving this into two forces, one (P,) perpendicular, the other (P^) parallel to BH, we have for the former and for the latter J^w^ .imp .m$ ^ 5 P,, 2 "" ^ * See a denxmstnetion at p. 309, vol. il. tenth edition of Dr. Hatton't ConiM of MatbcmiitiGt. 200 PRBS8URB OP EARTH AGAINST WALLS. [PART 1% ' Therefore tlie friction along the surface B H, will be fh^w^ . ton /9. sing y, . which subtracted from P^, gives (l_/.tang)*!i^l^=S (VII.) equal the strain acting through the centre of gravity of the prism^i- upon the back of the wall ; now, as the centre of gravity of a triangle^ is situate at a third of its height, and the direction of the strain S i^ parallel to the sloping base B H, it will meet the wall at one-third or" its height, and b being put for the breadth of the base of the wall, we shall have st^.sinp-b.coBp\ (VIII.) equal the moment of the strain S about the point A, tending to overthrow the wall. Example, Suppose a wall is to be built of brickwork (weighing 117 lbs. per cubic foot), to support a terrace 39 feet in height, the earth composing which weighs 105 lbs. per cubic foot, its natural slope being 53° from the vertical, and the coefficient of friction *45 ; it is required to find the breadth which must be given to the wall at its base in order to ensure its not being overthrown. By substituting these values in formula (VIIL), we obtain S = (1 - -45 X -4986) «^' X iO^ X '^^^^ = „630; the moment of which from formula ( VIII.) is 27636 f^ X -4462 - ^ x 8949^ = 160322 — 24731 . b. Therefore by formula (III.), we have ^^^^ "^ > 160322 - 24731 b, or b > 4-58 ; that is, the base of the wall must be more than 4*58 feet, in order that it shall not be overthrown by the pressure of the earth. 4. Of the experimentol results, the best which we have seen are those of M. Mayniel, from which the following are selected; in all of which the upper surface of the earth and of the wall which supports it are supposed to be both in one horizon tol plane. 1st. Both theory and experiment indicate that the resultant of the thrust of a bank, behind a vertical wall, is at a distance B K from the bottom of the wall equal to one-third of its height. 2dly. That the friction is half the pressure, in vegetoble earths, and four-tenths in sand; or tliaty=s *5 in one case and '4 in the other. .11.] PRBS8UBB OF BABTH AGAINST WALLS. 201 ly. The cohesion which vegetahle earths acquire, when cut in and well laid, course hy course, diminishes their thrust hy full kirdt; or in this case y = '667» 16 following tahle contains the value of the angle C B F, (heing the angle formed hy the natural slope and the vertical,) for al different kinds of earth, the authorities heing given in the last on. Nature of Earth. Angle Authority. Inbble Axme shingle perfectly dry 2arth the most dense and compact .^ommon earth tliffhtlv damp . Idem pulverized and dry Vegetahle earth Idem mixed with large g^vel Idem mixed with imall g^vel Sand 'Ine dry sand .... Idem Idem (a single experiment) 22 30 26 30 17 30 18 21 35 31 43 31 43 35 52 34 6 25 30 27 45 34 30 MaynieL Pasley. Barlow. Rondelet. Id. MaynieL Id. Id. Id. Barlow. Rondelet. Oadroy. THICKNESS OF WALLS, BOTH PACES VERTICAL. The following tahle exhibits the thickness which ought to he i to a parallel wall of various materials, and supporting the ore of different kinds of earth. Thickness of the wall or DC, its height being 1. Nature of Earth. WaUof brick. 100 Ibt. pereub. ft. WaU of un- hewn itooe, lasibs. per cab. ft. Wall of hewn flreotooe. taUe earth, carefully hud course by course , weU rammed li mixed with Uu-ge gravel •16 17 •19 •33 •15 •16 •17 •30 •13 •14 •16 •26 . Far waUs with an inUricr slope^ or a slope towards the hank, 16 hose of the slope he — of the height, and let S and « he the fie gravities of the wall and of the earth ; then DC =^ A/ -— -I- m ; V 3»* S n « m ^ •0424, for vegetahle or clayey earth, mixed -with large 202 BQUILIBBIUM OF P0LT€K)1I8. [PABT n. graTe] ; *0464>, if the earth be mixed with mall grmTel ; '1528, for sand; and '166, for semi-fluid earths. Example, Suppose the height of a wall to be 20 feet, and ^ of the height for the base of the tcdm9 or slope; suppose, also, the specific gravities of the wall and of the bank to be 2600, and 1400, and the earth semi-fluid; what, then, must be the thickness of the wall at the crown ? Here the theorem will become. ^^='^Vi^+-*^^-^^-^» = 20 >/ -0008333 + -0894 — 1 = (20 x '3) - 1 = 6 — 1 = 5 feet : while the thickness of the wall at bottom will be 6 feeL EQUILIBRIUM OF POLYGONS. 1. Let there be any number of lines, bars, or beams, AB, BC, CD, DE, &c. (fig. 175), all in the same vertical plane, connected together and freely moveable about the joints or angles. A, B, C, D, £, &c., and kept in equilibno by weights laid on the angles : it is required to assign the proportion of those weights ; as also the force or push in the direction of the said lines ; and the horizontal thrust at every angle. Through any point, as D, draw a vertical line a J} fig; to which, from any point, as C, draw lines in the direction of, or parallel to, the given lines or beams, viz., C a parallel to A B, Ch parallel to BC, Ce to DE, C/to EF, Cy to F O, &c. ; also CH parallel to the horizon, or perpendicular to the vertical line adg^ in which also all these parallels terminate. Then will all these lines be exactly proportional to the forces acting or exerted in the directions to which they are parallel, whether vertioil, honxontal, or oblique. That is, the oblique forces or thrusts in the direction of the bars AB, BC, CD, DE, £ F, F6, are proportional to their parallels ... Ca, C6, C</, C^, CJ\ Cg; and the vertical weights on the angles B, C, D, E, F, 9ic. are as the parts of the vertical a 5, &D, De, ej\ fg^ and Uie weight of the whole frame ABCDEFQ, is proportional to the sum of all the verticals, or to ag\ also the horizontal thrust at every angle, is everywhere the same constant quantity, and is expressed by the constant horizontal line CH. Cord, 1. It is worthy of remark that the lengths of the bars A B, B C, &c. do not affect or alter the proportions of any of these loads or thrusts ; since all the lines C a, C ^, a 6, &c., remun the same, whatever be the lengths of A B, B C, &c. The positions of the bars, and the weights on the angles depending mutually on each other, as well as the horizontal and oblique thrusts. Thus, if di^re be givea CBAP. n.] BQUILIBBIUM OF P0LT001V1. 203 the positioo of D C, and the weights or loads laid on the angles D, C« B; set these on the Tertical, DH, D6, ba^ then C&, Ca, give the direetiooB or positions of C B, B A, as well as the quantity or proportion of C H the constant horizontal thrnst. Vorol. S. If C H be made radios ; then it is evident that Ha is the tngent, aad C« the secant of the elevation of C a or A B ahove the horiaon ; also H 6 is %he tangent, md C b the secant of the elevation of C^ or CB ; also H D and CD the tangent and secant of the eleva- tion of C D ; also H e and C s the tangent and secant of the devadon of C« or D B ; also B/ and C/ the tangent and secant of the eleva- tion of EF; and so on; also the parts of the vertical a&, 6D, e/, /y, denoting the weights laid on the several angles, are the differences of the said tangents of elevations. Hence then in general, 1st. The oblique thmsts, in the directions of the bare, are to one another, directly in proportion to the secants of their angles with the borixon; or, which is the same thing, reciprocally proportional to the eosines of the same angles, or reciprocally proportional to the sines of the vertical angles, a, ^, D, e, /^ y, &c., made by the vertical line with the several directions of the bars; because (formula K . 3, page 137) the secants of any angles are always reciprocally in pro- portion to their cosines. 2. The weight or load laid on each angle is directly proportional to the difference between the tangents of the elevations above the boriaon, of the two lines vrhich form the angle. 3. The horiaontal thrust at every angle is the same constant quan- tity, and has the same proportion to Uie weight on the top of the q ppe raio st bar, as radius has to the taneent of the elevation of that W. Or, as the whole vertical a^ is to the line C H, so is the weight of the whole assemblage of bars, to the horizontal thrust. 4. It may hence be deduced also, that the weight or pressure laid on any angle, is directly proportional to the continual product of the sine of that angle and of the secants of the elevations of the bare or lines which form it. Sekokum. This proposition is very fruitful in its practical conse- quences, and contains the whole theory of centerings, and indeed of arches, which may be deduced from the premises by supposing the constituting bare to become very short, like arch stones, so as to fonn the curve of an arch. It appeara too, that the horizontal thrust, which is constant or uniformly the same throughout, is a proper memsuring unit, by means of which to estimate the other thrusts and pressures, as they are all determinable from it and the given posi- tions ; and the value of it, as appeare above, may be easily computed from the uppermost or vertical part alone, or from the whole assem- blage together, or from t^y part of the whole, counted from the top downwards. In the most Important cases, a model of the structure may be made, and the relations of the pressures at any angle, whether hori- zontal, ^erticsl, or in Qie directions of the beams, may be determined by a spring steel-yard applied successively in the several directions. 204 BQUILIBRIUM OP POLYGONS. [PABT II. 2. If the whole figure in the preceding prohlem be inverted, or turned round the horizontal line AG (fig. 176) as an axis, till it be completely reversed, or in the same vertical plane below the first position, each angle D, dy &c., being in the same plumb line; and if weights t, k^ /, m, n, which are respectively equal to the weights laid on the angles B, C, D, E, F, of the first figure, be now suspended by threads from the corresponding angles h^ e^ d^ d^ f^ of the lower figure; those weights keep this figure in exact equilibrio, the same as the former, and all the tensions or forces in the latter case, whether vertical, horizontal, or oblique, will be exactly equal to the corresponding forces of weight, pressure, or thrust in the like directions of the first figure. This, again, is a proposition most fertile in its application, especi- ally to the practical mechanic, saving the labour of tedious calcula- tions, but making the results of experiment equally accurate. It may thus be applied to the practical determination of arches for bridges, with any proposed roadway ; and to that of the position of the rafters in a curb or mansard roof. 3. Thus, suppose it were required to make such a roof, with a given width AE {^g, 177), and of four proposed rafters AB, BC, CD, DE. Here, take four pieces that are equal or in the same given proportions as those proposed, and connect them closely toge- ther at the joints A, B, C, D, E, by pins or strings, so as to be freely moveable about them; then suspend the whole from two pins, A, E, fixed in the same horizontal line, and the several pieces will arrange themselves in such a form, A^c^/E, that all its parts will come to rest in eauilibrio. Then, by inverting the figure, it will exhibit the form of the framing of a curb roof A B C D E, which will also be in equilibrio, the thrusts of the pieces now balancing each other, in the same manner as was done by the mutual tensions of the hanging festoon A6c</£. 4. If the mansard be constituted of four equal rafters; then, if angle CAE = m, angle CAB=:;i?; it is demonstrable that 2 sin 2 ;r = sin 2 m. So that if the span A E, and height M C, be given, it will be easy to compute the lengths AB, BC, &c. EwampU. Suppose A E = 24 feet, M C 12. IMF C* Then, —— = 1 = tan 45° angle C A M = m. MA .'. sin 2 ??» 1= sin 90° = 1, and sin 2 ;r = ^ .-. 2a; = 30°, and « = 15° =i CAB. Hence M A B = 45° -f 15° = 60° and M B A = J (180° — 2 x 15°) = 90° — 15° = 75° also A M B = 180° — (75° -h 60°) = 45° and lastly, sin 75° : sin 45 : : AM = 12 : AB = 8*7846 feet Note. — In this example, since AM = MC, as well as AB = BC, it is evident that AI B bisects the right anele A M C ; yet it seemed nreferable to trace the steps of a general solution. CHAP. II,] STABILITY OF ABCHBS. 205 STABILITY OF ABCHBS. 1 . If the effect of the force of gravity upon the ponderating matter of an arch and pier, he considered apart from the operation of the cements which unite the stones, &c., the investigation is difficult to practical men, and it furnishes results that require much skill and care in their application. But, in an arch whose component parts are united with a veir powerful cement, those parts do not give way in vertical columns, hut hy the separation of the ientire mass (including arches and piers), into three, or, at most, into four parts ; and in this case the conditions of equilibrium are easily expressed and applied. LetyF, /'F', (fig. 178) be the joints of rupture, or places at which the arch would most naturally separate, whether it yield in two pieces or in one. Let G be the centre of gravity of the semi- arch /F K *, and G' that of the pier A B F/. Let 'f I be drawn parallel to the horizon, and G H be demitted perpendicularly upon It ; also let G^ D be a perpendicular passing through G^, and F £ diBwn from F parallel to it. Then, 2. The first case is when the arch fY Y' f tends to fall vertically in one piece, removing the sections J^F, f'Y'\ if W be the weight of the semi-arch fY K k^ and P that of the pier up to the joint y F, the equilibrium will be determined by these two equations : — viz., * • P = w 0? - *) (I.) ."=-(n-^0 ("■' where ^ is the measure of the friction, or the tangent of the angle of repose of the material, and the first equation is that of the equili- Imnm of the horizontal thrusts, while the second indicates the equili- lirium of rotation about the exterior angle A of the pier. 3. In the second case, when each of the two semi-arches Yk^k F\ tend to turn about the vertex k of the arch, removing the points F, F', the equilibrium of horizontal translation, and of rotation, will he respectively determined by the following equations : — viz., * . p = w (j^-i) (HI.) 'H-C^-ID- (-, 4. Hence it will be easy to examine the stability of any cemented arch, upon the hypothesis of these two propositions. Assume dif- ferent points, such as F in the arch, for which let the numerical values of the equations (I.) and (11.), or (III.) and (IV.) be com- puted. To ensure stability, the first members of the respective eaoations must exceed the second ; and those parts will be weakest where the excess is least A F£ 206 ARCHBS AND PIBB8. [part 5. The following table extracted from Prof. Moseley's ^^ Mechanical Principles of Engineering," page 151, contains the value of the angle ^ for the materials most usually employed in the construction of arches. Nature of Materials. GoeffidcBt of firiction. Lumiting Soft calcareous stone, well dressed, upon the same . Hard calcareous stone, ditto rommon brick, ditto Oak, endwise, ditto Wrought iron, ditto Hard calcareous stone, well dressed, upon hard cal- I careous stone I 8oft, ditto Conamon brick^ ditto Oak, endwise, ditto Wrought iron, ditto Soft calcareous stone upon soft calcareous stone, with 1 fresh mortar of fine sand ) EZPERTUENTS BT DIFFERENT OBSERVERS. Smooth freestone upon smooth freestone, dry. > (Rennie.) y Ditto, with fresh mortar. (Rennie.) Hard polished calcareous stone upon hard polished } calcareous stone • . . . . S Calcareous stone upon ditto, both surfaces being made \ rough with the chisel. (Bouchard!.) . . > Well dressed granite upon rough granite. (Rennie.) Ditto, with fresh mortar, ditto. (Rennie.) Box of wood upon pavement. (Ili^ier.) Ditto, upon {>eaten earth. (Herbert.) Libage stone upon a bed of dry clay Ditto, the clay being damp and soft Ditto, the clay being equally damp, but covered with 7 thick sand. (Oreve.) ) 74 •76 •67 •63 •49 •70 •75 •67 •64 •42 •74 •71 •68 •78 •66 •49 •68 •33 •61 •34 •40 36 30 36 62 33 50 32 13 26 35 36 62 33 60 32 37 22 47 36 SO 36 23 33 26 30 7 37 68 33 26 26 7 80 7 18 16 27 2 18 47 21 48 If the section be drawn on smooth drawing pastehoard, upon a good sized scale, the places of the centres of gravity may he found experimentally, as well as the relative weights of the semi-arch and piers, and the measures of the several lines from the scale employed in the construction. If the dimensions of the arch were given, and the thickness of the pier required ; the same equations would serve ; and different thick- nesses of the pier might he assumed, until the first members of the equations come out largest. The same rules are applicable to domes, simply taking the un- gulas* instead of t^e profiles. * The ungulas^ as mentioned above, are the solids generated by the revolu- tion of the sections /FirK and /FA B about the vertical axis ArC. CflAP. II.] 8U8PENM0II BBIDOBB. SOT BQUILIBBIUH OF SUSPENSION BRIDGES. 1. The several parts of a suspension bridge should always be pro- portioned in such a manner, that the tensile strain per square inch of section would be nnifbrm throughout the whole length of the chains ; and as the tension varies with the inclination of the chain, and to fulfil the above condition, the area of the chain must vary as the tension, it is q«ite evident that the chains of a suspension bridge ought not under any circumstances to be made of a uniform sectional area. If the chain was of a uniform section, and had only its own weight to carry, the form which it would assume would be a catenary; if on the other hand, the chain is supposed devoid of weight, and the load suspended fhnn it to be uniformly distributed horixontally, (as in the case of the roadway of a bridge,) then the form which the chain would assume would be the common parabola. Neither, however, of these supposed cases ever occur in practice, there being always three loads very differently distributed, viz., the weight of the chain itself, that of the roadway suspended from it, and that of the vertical rods by which the same is suspended ; consequently, the form which the chain when in equilibrium would assume is neither a catenary, nor a common parabola, but is between the two; and it approaches nearer to one or the other, as the weight of the chain or that of the roadvray predominates. Taking these three loads into accoun^ and varying the section of the chain so that its tensile strain is uniform tbroashout, the determination of the true form of the curve which it vodd assume when all its parts were in equilibrium, becomes a very eoiDplicated problem; it Ims, however, been very ably solved by Professor Moseley*, and the following formula which he has de- daoed, contains aU that is required for determinmg the form of the eorre end the dimensions of the chains. If fi^ = the weight of a bar of the material of the chain 1 square inch in section and 1 foot long, /a, :s the weight of a foot in length of the roadway, and, supposing the vertical suspending rods to be diflfbsed over the whole space between the chain and the road- ]Bray, formins a uniform piate of such a thickness that its weight is precisely uie same as that of the actual suspending rods, let ^13 = the "weight of a square foot of this plate ; also, let K := the sectional %Tea in square inches of the chain at any point P (fig. 179), x and y being the ordinate and abscissa at the same point; let a = the Bemi-span, H = the deflexion, b = the length of the shortest sus- pending rod, € =3 the tension upon the lowest point of the chain, mMkd m c3 — ^^ where r represents the tenacity of the material of T Che chain per square inch, and m = the number of times that r exceeds the actual tensile strain upon the chains ; then * ** The Mechanical Principles of Engineering and Architecture,*' by the Her. Henry Af oaeley, page 647* 208 SUSPENSION BBID0B8. [PART II. - = C.>-e'.Uj »-') <■•) "(iriir^y ("•) K _^^|2(,_ j)(!ii±!U + .) + ,}'.. (II,.) 2. As, however, the use of these formulsB for determining the requisite numher of points in a hridge would be attended with much labour, it is only in cases where great accuracy is requisite that they need be employed ; for most purposes it will be sufficiently correct to assume that the form of the curve is a common parabola, in which case the formulse for determining the several elements of the bridge become much more simple, and easy of application. Let t ^ the coefficient of tension at the lowest point of the chain, or the quantity by which half the weight of the bridge must be multiplied to give the actual strain upon the chains, t^ = the coeffi- cient at the point of suspension, t^ := the mean coefficient of tension for the whole chain, « := the semi-span, d = the deflexion, x = the leoeth of half the chain, K^ = the sectional area of the chains at the lowest point, K^ = the same at the point of suspension, K3 := the mean sectional area of the whole chain, W := the weight of half the whole bridge, including the chain, y = the angle which the tangent to the curve at the point of suspension makes with the horizon, and b^ = the length of the suspending rod at any point P. The letters, /a^, /a^,, m, t, ^, :r, ^, and c, represent the same quan- tities as before. Then, the curve being a parabola, the length of the suspending rod at any point P may easily be found by means of the following property of the parabola ; y : rf : : a?' : «, therefore, supposing the roadway to be horizontal, we have h, =^ + V^ (IV.) And for the length of the chain, we have from the formula at page 129, for the length of parabolic arcs. =V •* + |rf' (V.) Also, from the method given at page 174, for drawing a tangent to any point in a parabola, we can easily deduce the value of 7, as follows: — if DE (fig. 180) be made equal to C D, the line drawn from A to £ will be the tangent fo the parabola at the point A, and therefore the angle CAE = 7, is determined from the proportion B : rad : : 2d : tan 7; CHAf. II.] SUSPENSION BRIDGES. 209 .. tany = — (VI.) ^iid sioce the chain A D B with the roadway suspended from it ^ notbiog more than an equilibriated polygon, such as is described at page 202, having an infinite number of sides, all the relations there mentioned as existing between the several strains, may be applied to the present inquiry; therefore, if CE = 2rf be made to "^Pwsent the whole weight of half the bridge = W, then A C s= « ^^^ represent the tension at D, and A E the tension at A ; therefore, ''=2^ (V") ^ = '-^ (VIII.) •nd <, = (I + «sc>) ^ (IX.) lien, for the mean sectional area of the chain, we have K,= ^^»' (X.) •"^d for the weight of the roadway and chain for half the bridge, W = K3X^j+fi2« (XL) *Tjen since /j W = c, the tension at the lowest part of the chain, ^^ have for the sectional area of the chain at the same place, K, =^!lZ (XII.) T ^^ K W being the tension at the point of suspension, the sectional **^ of the chain at that point will be K, =^^ ^Xjjj^ ^ ^* In order to render the practical application of the foregoing ^i^nle quite clear, we subjoin the following example. Let it be '^iriTed to determine all the elements of a suspension bridge, tlie •P*Ji of which is to be 360 feet, and the deflexion 30 feet; let the y^^} of the roadway for everj' foot in length = 4500 lbs., the ^^**ciiy of a square inch of the chain = 67200 lbs., the weight of a ^ 1 inch square and 1 foot long := 3*4 lbs., and the number of r^ that the cohesive strength of the iron is to exceed the "^ mbstituting these values for the letters representing the ^«il qumiities, we obtain from formula (V.) p 210 SUSPENSION BRIDGES. [Pi 4 X SO'* 180 + — = 183-33 feet, equal the length c the chain. From formula (VI.) 2 X 30 1 tan V = = - = -3333, which is the tangent* of li '^ 180 3 ' ^ the angle made hy the chains with the horizon at their pc suspension. From formula (VII.) 180 t, = = 3, the coefficient of tension at the lowest i * 2 X 30 the chain; from formula (VIII.) 180 X 10541 1 „,^^„ , ,, «• * r* / = = 3*1623 equal the coefficient of teui 2 2 X 30 ^ the point of suspension ; and from formula (IX.) 180 f, =(1 -f 1-0541) -. =30812 for the mean coeffici tension for the whole length of the chain. From formula (X.) 4500 X 180 67200 — 1— — - - 183-33 X 3-4 6 X 3-0812 mean sectional area of the chain. From formula (XL) W = 269 X 183-33 x 3;4 + 4500 X 180 = 977,647 lbs. \ total weight of half the bridge; and finally, from formula (XI obtain 6 X 3 X 977,674 ^^^ • u r *a. Kj = =262 square mches for the section of the chain in the centre ; and from formula (XIII.) ex 3-1623 X 977,674 ^^^ • t, r .u Kp = ^^^^^ = 276 square inches for the se * 67,200 ^ areA of the chain at the point of suspension. "• In order to find the angle of which '3333 ii the tangent, take tl '3333 a 7*522835, and adding 10 to the characteristic (for the reason < which see page 153), we have 9*522835, which we find from Table IV log tan of IS"" 26^, as above. t The secant of 18** 26' is obtained by a reverse process to that ezpl the foregoing note ; thus, from Table IV. we find the log sec of W\ 10*022875, then subtracting 10 we have 0*022875, the number answ which in Table III. is 1 0541, the secant of 18** 26^ radius being 1. K3 = cpfa/\t\ ' — *^~ ^ ^^^ square inches f CHAP, in.] GBNBRAL DBFINITIONS. 211 CHAP. III. DYNAMICS. Sect. I. General Dejtnitions, 1. The man of a body is the q^uantity of matter of which it ift «wnp08ed; and is proportional to its weight, or to the^brc^ which nost be applied to the body to prevent its gravitating to the earth, i&d wbicb, being greater or less as the mass is greater or less, we i^u a measure of the mass itself. 3. Demit^ is a word by which we indicate the comparative close- ^ or otherwise of the particles of bodies, and is synonymous with ^ term tpeeifie gravity. Those bodies which have the greatest num- ^ of particles, or the greatest quantity of matter, in a given magni- ^C) we call moA dense; those which have the least quantity of ^*il(tt, leatt dejtse. Thus lead is more dense than freestone ; yre«- ^^ more dense than oak ; and oak more dense than cork. . 3. The velocity with which a body in motion moves, is measured ^y the space over which it passes in any given time; the unit usually ^^Mned being one second. 4. If the body passes over an equal space in each successive unit ^ time, the body is said to move uniformly^ br to have a uniform ^^ki^y and the measure of such velocity is the space actually passed ^^er b? the body in each second. S* If, however, the body passes over a yreafer space in each sue- ^^▼e second tlwn it did in the preceding, then it is said to move ^tt ao aeceleraied vdocUy; when the differences between the spaces ^^^ oter in any two successive seconds is the same, at whatever l^^nod of the body's motion they be taken, or in other words, when ^ HKceaaive spaces form an arithmetical progression, the body is ^^ to move with a uniformly accelerated velocity ; but when the ^itts passed over in successive seconds increase according to any ^^ law, the body is then said to have its velocity variably accele- ^ If^ on the other hand, the body passes over a smaller space in ^ SQoeeasive second than it did in the preceding, then it is said to ^e with a retarded velocity; which, if the successive spaces form a ^<c<eaaiig arithmetical series, is said to be uniformly retarded; if ^^kerwiae, it is said to be variably retarded, p 2 212 LAWS OF UNIFORM AND VARIABLE MOTION. [PART II. 7. The velocity of a body whose motion is Tariable is expressed at any moment, by the space which it tcould pa^s ocer in a second^ if its Telocity at the moment spoken of were to continue uniform for that period. 8. Mechanical effect is measured by the product of the nuus or weight of the body into the space over which it has been moved; no regard being had to the time occupied. The unit of mechanical effect employed in the subsequent pages, is a weight of 1 pound raised through a space of 1 foot, and is designated by the letter U. 0. The momentum of a body in motion means the mechanical effect which such a body will produce in a moment (or second) of time, and varies as the weight of tlie body multiplied by its velocity, 10. The vis viva of a body in motion is the whole mechanical effect which it will produce in being brought to a state of rest^ no regard being had to the time in which the effect is produced, and it varies as the weight of the body multiplied by the square of its vel4>city. Sect. II. On the general Latcs of Uniform and Variable Motion. 1. As a proper understanding of the actual difference between the momentum and the vis viva of a body in motion, is esjtential to a correct application of the principles of dynamics, we shall take some pains to set this difference in as clear a light (for the student) as pos- sible. And it is of the more importance to do so, ns a diversity of opinion upon this subject has existed amongst some of the most emi- nent mathematicians, and much time and talent has been expended by them in supporting errors w Inch have arisen entirely in a miscon- ception of terms, ai)d in excluding from their conclusion the con- sideration of time which they included in their premises. 2. From carefully conducted and often repeated experiments, the following results with regard to bodies in motion have been ob- tained : — I. If a body of a certain weight, and moving with a given velocity, meet another body of double that weight, and moving with half the velocity, the two bodies will destroy each other's motion, and both will be brought to a state of rest. II. A body of a certain weight and moving with a given velo- city, being subject to a uniformly retarding forc^, (i . e, a uni- form force acting constantly in a contrary direction to the body's motion,) will move over a certain space in being brought to rest, and will occupy a certain time in doing so ; then another body of the same weight, but moving with half the velocity of the former, being Subject to the same uni- formly retarding force, will move over one quarter of the CHAP. III.] LAWS OF UNIFORM AND VARIABLB MOTION. 213 space moTed over by the former, in being brought to a state of rest, and will occupy in doing so ^o^tbe time. And an- other body of the same weight, but moving with one-third of the velocity of the first, will move over one-ninth of the space, and occupy one-third the time of the first, in being brought to a state of rest. Now the diversity of opinion to which we have alluded above, has arisen from the (at first sight) apparently contradictory nature of these two results : one party has drawn a conclusion from the first experiment that the force of a body in ynotion is directly <w its velo- city; and the other party has drawn a conclusion from the second experiment that \he force of a body in motion is directly as the square of its velocity. These errors have arisep from the term " force of a body in motion" being used without any fixed and definite meaning being attached to the same. 3. The proper measure of the whole force (i. e. vis viva) of a body in motion, is the mechanical effect, or (as Professor Moseley very significantly terms it) the work^ which it is capable of performing in being brought to a state of rest. That evidently being the force due to its faction which is required to destroy the same^ and which is directly as the square of its velocity^ as found by the second experi- ment. 4. If, however, we only consider the mechanical effect (or the work) which a body in motion is capable of performing in a given time, (i.e. its momentum^) we shall find from the second experiment, that ajthough the body having twice the velocity ultimately produced four times Qie effect, in doing so it occupied just twice the tii>c; and ^in, although the body moving with three times the velocity tUti- niatefy produced nine times the effect, it occupied in doing so three times the time ; and, therefore, the mechanical effect produced in a yicen time by the bodies was directly as tlie velocities. And this con- clusion, drawn from the second experiment, is in accordance with that which must be drawn from the first experiment, in which it is eyident that the effect produced by both bodies must be in the same (although indefinitely small) time. 5. The force of gravity being constantly the same, both in amount %iid direction, and being practically uniform* in its action for such ^niall distances from the earth's surface as come under consideration iQ ordinary dynamical investigations, has been universally adopted as the onit of measure for all other forces. 6. The actual amount of the force of gravity is measured by the Velocity which a body falling in vacuo, in the latitude of London, will * The force of gravity varies as the square of the distance from the earth*s oencre, and therefore becomes less as we asueiid alxive the surface of the earth, in the proportion of I to *9994, a difference too small to require notice in any question of terrestrial mechanics. 214 MOTION UNIFORMLY ACCBLBBATED. [PABT II. acqaiie in one second of time, and which, hy carefully conducted ex- peiiments, has heen ascertained to he 386*289 inches, or about 32^ feet per second^ and this latter quantity is usually represented by the symbol g. MOTION UNIFORMLY ACCBLERATED. 1 . If we now put W to denote the quantity of matter, or weigLt of a body in motion, v its velocity in feet per second, M its momen- tum, and V its vU viva^ both expressed in units of mechanical effect, or pounds raised through a space of 1 foot; we have M = Wfj (I.) ^ = ^2-^ <"•> 2. And further, if s h? put to represent the space passed over in the time ^, by a body subject to the uniform forced*; then we have the following relations between all these quantities : — V = «W = -W=-yW = -2/W . (III.) 2 22/ ^^'> «=//=—= ^2fB .... (VI.) ' = ) = v-a/7 •• <™> MOTION OF BODIES UNDER THE ACTION OF GRAVITY. 1. When the uniform force is that of gravity,/ := g; and g being proportional to the weight of the body (W), the foregoing relations are simplified, and then become as follows : — , = ^ = ^J! = ^ (IX.) 2 2 2^ ^ ^ V^gt^— ^^/Yf% (X.) CHAP. III.] DE8CBNTS BY OBAVITV. 215 = • = — = /u (XI.) -=r-7' = n '-■) 2. Any two of these quantities being given, the other two may be immediately ascertained from the above equations; the following table shows their actual numerical values for the first four seconds of the motion of a heavy body filling freely by the action of gravity : — The times in seconds being 1", 2", 3", 4", &c. The velocities in feet will be 32^, 64 j, 96i, 128^, &c. The spaces in the whole times l6-j^^ 64|, 144}, 257|, &c. And the space for each second 16^, 48^, 80j^, 112^*^, &c. of which spaces the common difference is 32^ feet, equal y, the mea- sure of the force of gravity. 3. If, instead of a heavy bodv being allowed to fall freely, it be propelled vertically upwards or downwards with a given velocity, v, then M^tvzfi^; (XIII.) aa ezpressioD in which the upper sign — must be taken when the projection is upwards^ the lower sign + when the projection is down- wardM. 4. When only an approximate result is required with reference to bodies falling vertically, 32 may be put for ^, instead of 32 J : there would then result, in motions from quiescence, '-'"■-a"? • ■ • ■ (»"'•) t;=8v^i=: — = 32^ (XVI.) ^lius, if the space descended were 64 feet, we should have i; := 8 Q ^ B = 64 feet per second, and ^ = - = 3 seconds. If the space descended were 400; then t? = 8 x 20 = 160 feet l-^,e«o.d,«.d. = ¥ = 6eeco»dB. 4 5- The force of gravity differs a little at different latitiules ; the law ^ the variation is not as yet precUely ascertained ; but the following 216 MOTION OVEB PULLEYS, ETC. [PART II. theorems are known to represent it very nearly. That is, if ^ denote the force of grayity at latitude 46% ^j the force at the poles, ^^ the force at the equator, and ^3 the force at any other place : then ^, =^(1 + -002837) (XVII.) g^z=g{\ — 00-2837) (XVIII.) g^^g (1- — -002837 cos 2 lat.) . . (XIX.) MOTION OVER A FIXED PULLEY. In this case let the two weights which are connected hy the cord that goes over the pulley be denoted by W . and W , : then — ^ — — ^ 9 = W/in formula (III.); so that W,-W, gj_ W, +W, ' 2 Or, if the resistance caused by the rigidity of the rope, and the fric- tion and inertia of the pulley, be represented by r; then '-W,+W, + r • 2 ^^^-^ Example 1. Suppose the two weights to be 6 and 3 lbs. te&Y*^^' ively, what will be the space descended in 4 seconds ? 16 = 16jV X 4 = 64jfeet. Example 2. But suppose that, in an actual experiment witl» ^ weights of 6 and 3 lbs. over a pulley, the heavier weight desce-^^ ^^ only 50 feet in 4 seconds. the same in both examples, we have w, + W^ + r : W. + W, :: 64J : 60 or, dividendo r : W, -f W^ :: 14J *: 60 that is, r : 5 -f 3 :: 14J : 60 whence r = ^^tJ^ = ^4^^ = 22933 lbs. 50 50 the measure of the resistance and the inertia. 2. Similar principles are applicable in a variety of other caies / and by varying the relations of W„ W^, and r, the force may hvre CHAP. III.] MOTION ON INCLINED PLANES. 21? any assigned ratio to that of gravity; which is, indeed, the founda- tion of Mr. Atwood's elegant apparatus for experiments on accele- rating forces; an account of which may he seen in the 2nd volume of my Mechanics^ or in almost any of the general dictionaries of arts and sciences. 3. If, instead of pulleys, small wheels and axles, as in figure 181, be employed^ to raise weights by the preponderance of equal weights: then, if the diameter of the wheel and axle A be as 3 to 2 ; those of the wheel and axle B, as 5 to 2; and those of C, as 8 to 2; it will be found that the weight b will be elevated more rapidly than either a or c : the proportion of 5 to 2, (or, more accurately, of 1 -f- ^^2 to I) being in that respect the most favourable. MOTION ON INCLINED PLANES. 1 . When bodies move down inclined planes, the accelerating force (independently of the modification occasioned by the position of the centre of gyration) is equal to g multiplied by the quotient of the beight of the plane divided by its length, or what is equivalent, by the sine of the inclination of the plane. In this case, therefore, put- ting « for the inclination of the plane, the formulae become (XXI.) _ ^ ^ . sin I ^ tv 2 2^ . sin f "" 2 2 9 r = y / . sin I = >/ {^g9 . sin t) = — (XXII.) *=./-^'-=^' (xxm.) Purther, if », be the velocity with which a body is projected up or "Oivn a plane, then c = Vj ip y / . sin I (XXIV.) ^^ . sin f v^ — v^ , s=^v^t^^ =-~^ —. . . (XXV.) * ^ 2 2^ . siuf ^ ^ Alaking v, = 0. in equation (XXIV.), and the latter' member of ^^ Nation (XXV.), the first vnll give the lime at which the body will '^^'^^e to rise, the latter the spctce. Example. Suppose a body be projected up a smooth inclined p^Be whose height is 12 and length 193 feet, with a velocity of 20 .^^t per second, how high will it rise up the plane before its motion ^^ extinguished? Then substituting these values in formula (^XXV.), wc have 400-0 400 400 ,_^ , . , "= 64J X ^»/, =193 12 =-^ = 100fcet,thcspacerequired. 3 ^ 193 218 MOTION ON INCLINED PLANES. [PART II. 2. With regard to the velocities acquired by bodies in falling down planes of the same height, this proposition holds ; viz. that they are all eaual, estimated in their respective directions. Thus, if AD, BE, CF, (fig. 182), be planes of different inclinations, and AC, DF, hori* zontal lines, the balls A, B, C, after descending along those planes will have equal velocities when they arrive at the points D, £, F, re- spectively. 3. Also, all the chords, such as AD, BD, CD, (fig. 183), that ter- minate either in the upper or the lower extremity of the vertical dia- meter of a circle, will be described in the same time by heavy bodies A, B, C, running down them; and that time will be equal to die time of vertical descent through the diameter DE. 4. If three weights, asA, B, C, (fig. 184), be drawn up three planes of different inclinations, by three equal weights hanging from cords over pulleys at P, then if the length of the middle plane be ti0ice its height^ tLe body B will be drawn up that plane, quicker than either of the other weights A or C. Or, generally, to ensure an ascent up a plane in the least time, the length of the plane must be to its heighty as ttoice the weight to the power employed. 5. If it be proposed to construct a roof over a building of a given width, so that the rain shall run quickest off it, then each side of the roof must be inclined 45° to the horizon, or the angle at the ridge must be a right angle. 6. The force by which spheres, cylinders, &c. are caused to re- volve as they move down an inclined plane (instead of sliding) is the adhesion of their surfaces occasioned by the pressure against the plane : this pressure is part of the body's weight ; for the weight being resolved into its components, one in the direction of the plane, and the other perpendicular to it, the latter is the force of the pres- sure upon the plane; and, while the same body rolls down the plane, will be expressed by the cosine of the plane's elevation. Hence, since the cosine decreases while the arc or angle increases, after the angle of elevation ariives at a certain magnitude, the adhesion may become less than what is necessary to make the circumference of the body revolve fast enough ; in this case the body descends partly by sliding and partly by rolling. And the same may happen in smaller elevations, if the body and plane are very smooth. But at all eleva- tions the body may be made to roll by Uie uncoiling of a thread or riband wound about it. If W denote the weight of a body, a the space described by a body falling freely by the action of gravity, or sliding freely down an in- clined plane, then the spaces («,) described by rotation in the same time by the following bodies, will be in these proportions. (1.) In a hollow cylinder, or cylindrical surface, «| = ^ « and the tension of the cord, in the first case = | W. (2.) In a solid cylinder, «, = ^ «, and the tension = ^ W. (3.) In a spheric surface, or thin spherical shell, «, = |^ «, and the tension = | W. (4.) In a solid sphere, «| =: -^ «, and the tension = -^ W. CHAP, in.] MOTION ON INCLINBD PLANES. 210 If two cylinders be taken of equal size and weight, and with equal protuberances upon which to roll, as in the mar- ginal figures: then, if lead be coiled uniformly over the curve surface of B, and an equal quan- tity of lead be placed uniformly from one* end to the other near the axis in the cylinder A, that cylinder will roll down any inclined plane quicker than the other cylinder B. The reason is that each partide of matter in a roUing body reiisU motion in proportion to the squarb of its distance JrofH ike axis of motion ; and the particles of lead which most resist motion are placed at a greater distance from the axis in the cylinder B than in A. 7. The friction between the surface of any body and a plane, may be eanly ascertained by gradually elevating the plane untiJ the body upon it/iM/ begins to dide. The friction of the body is to its weight as the height of the plane is to its base, or as the tangent of the in- clination of the plane is to the radius. Thus, if a piece of stone in -weight 8 pounds, just begins to slide when the height of the plane is 2 feet, and its baise 2\ ; then the friction will be ^ the weight, or 4 of 8 lbs. = 6| lbs. 8. After motion has commenced upon an inclined plane, the fric- tion is osoally much diminished. It may easily be ascertained expe- Timentaiiy, by comparing the dme occupied by a body in sliding down a plane of given height and length, or given inclmation, with that which the simple theorem for /, (XXIII.) would give. For, iff he the value of the friction in terms of the pressure, the theorem for the time wiU be ^1 = A / — T-' "^y instead off = a / : — • Hence r,«:<«::sini:sini-/ . . . (XXVI.) Example. Suppose that a body slides down a plane in length 30 feet, height 10, in 2^ seconds, what is the amount of the friction? Here f « A / ^ = A /;;7n r = 2-366 nearly. Hence (2-6)» : (2-366)' :: § : -27603 = sin • -/ Consequently, -33333 — 27603 = -0573 value of the friction, the weight being unity. 9. When a weight is to be moved either up an inclined plane, or along on horizontal plane, the angle of traction PWB (fig. 185) that the weight may be drawn with least effort, will vary with the value off. The magnitude of that angle PWB for several values of f are exhibited in the following table: — 220 MOTIONS ABOUT A PIXBD CBNTBR OB AXIS. [PABT IT. / PWB 1/ PWB / PWB / PWB / PWB ij/ PWB 1 45° 0' II 26°34' ^ 18^26' i 14° 2' t 11°19! ^ 9^28' t 38 40 23 58 ^ 16 54 tV 13 15 i\ 10 47 IJ4 8 8 ^ 33 41 ^ 21 48 -f 15 57 ^ 12 32 A 10 18 4 7 8 + 29 45 A 19 59 -h 14 56 A 11 63 A 9 52 i 6 20 10. If, instead of seeking the line of traction so that the moving force should he a nainimuro, we required the position such that the suspending force to keep a load from descending should he a mini- mum, or a given force should oppose motion with the greatest energy; then the angles in the preceding tahle will he still applicahlc, only the angle in any assigned case must he taken helow, as BWp. This will serve in the huilding and securing walls, in the construction of hanks of earth, fortifications, &c., and in arranging the position of land-ties^ &c. Sect. III. Motions abotU a fixed Center or Axis. Centers of Oscillation and Percussion. 1. The center ofosciUation is that point in the axis of suspension of a vihrating hody in which, if all the matter of the system were collected, any force applied there would generate the same angular velocity in a given time as the same force at the center of gravity, the parts of the system revolving in their respective places. Or, since the force of gravity upon the whole hody may be con- sidered as a single force (equivalent to the weight of the hody) applied at its center of gravity, the center of oscillation is that point in a vihrating hody in which, if the whole were concentrated and attached to the same axis of motion, it would then vihrate in the same time that the hody does in its natural state. 2. From the first definition it follows that the center of oscillation is situated in a right line passing through the center of gravity, and perpendicular to the axis of motion. It is always farther from the point of suspension than the center of gravity. 3. The center of percussion is that point in a hody revolving about an axis, at which, if it struck an immovable obstacle, the whole of its motion would be destroyed, or it would not incline either way. 4. When an oscillating body vibrates with a given angular velo- city, and strikes an obstacle, the effect of the impact will be the greatest if it be made at the center of percussion. For, in this case the obstacle receives the whole revolving motion of the body; whereas, if the blow be struck in any other point, a part of the motion of the body will he employed in endeavouring to continue the rotation. 5. If a body revolving on an axis strike an immovable obstacle CHAP. III.] PENDULUMS. 221 at the center of percussion, the point of suspension will not he affected hy the stroke. We can ascertain this property of the point of suspension when we give a smart hlow with a stick. If we give it a motion round the joint of the wrist only, and, holding it at one extremity, strike smartly with a point considerahly nearer or more remote than ^ of its length, we feel a painful wrench in the hand : but if we strike with that point which is precisely at ^ of the length (that being the situation of the center of percussion), no such dis- agreeable strain will be felt. If we strike the blow with one end of the stick, we must make its center of motion at ^ of its length from the other end; and then the wrench will be avoided. 6. The distance of the center of percussion from the axis of motion is equal to the distance of the center of oscillation from the same : supposing that the center of percussion is required in a plane passing through the axis of motion and the center of gravity. SIMPLE AND COMPOUND PENDULUMS. 1 . A simple pendulum^ theoretically considered, is a single weight, regarded as a point, or as a very small globe hanging at the lower extremity of an inflexible right line, void of weight, and suspended from a fixed point or center, about which it oscillates. 2. A compound pendulum is one that consists of several weights movable about one common center of motion, but so connected together as to retain the same distance both from one another and from the center about which they vibrate. Or any body, as a cone, a cylinder, or of any shape, whether regular or irregular, so suspended as to be capable of vibrating, may be regarded as a compound pendulum ; and the distance of its center of oscillation from any assumed point of suspension, is considered as the length of an equivalent simple pendulum. 3. If O represent the distance of the center of oscillation from the point of suspension, and G the distance of the center of gravity from the same point, it has been found that 6 O is a constant quantity, for the same body and the same plane of vibration. 4. Any such vibrating body will have as many cefiters of oscillation as yon give it points of suspension: but when any one of those centers of oscillation is determined, either by theory or experiment, the rest are easily found by means of the foregoing property that O G is a constant product, or of the same value for the same body. 5. When a body either revolves about an axis, or oscillates, the sum of the products of each of the material elements, or particles of that body, into the squares of their respective distances from the axis of rotation, is called the momentum of inertia of that body. 6. A point, or very small body, on descending along the successive sides of a polygon in a vertical plane, loses at each angle a part of its actual velocity equal to the product of that velocity into the versed sine of the angle made by the side which it has just quitted, and the 222 PENDULUMS. [part II. prolongation of the side upon which it is just entering. Therefore, that loss is indefinitely small in curves, 7. A heavy body which descends by the force of gravity along a curve situated in a vertical plane, has, in anv point whatever, the same velocity as it would have had if it had fallen through a vertical line equal to that between the top and the bottom of the arc run over : and when it has arrived at the bottom of any such curve, if there be another branch either similar or dissimilar, rising on the opposite side, the body will rise along that branch (apart from the consideration of friction) until it has reached the horizontal plane from which it first set out. Thus, after having descended from A to V (fig. 186), it will have the same velocity as that acquired by falling through D V, and it will ascend up the opposite branch until it arrives at B. 8. If the body describe a curve by a pendulous motion, the same property will be found to obtain, setting aside the effects of friction. Thus, let a ball hang by a flexible cord SD (fig. 187) from a pin at S : then, after it has descended through the arc DE, it will pass through an equal and similar arc £A, going up to A in the same horizontal line with D, and ascending from £ to A in an interval of time equal to that which it descended from D to £. But, if a pin projecting from P or p stop the cord in its course, the ball will still rise to B or to C, in the same horizontal line with A and D ; but will describe the ascending portions of the curve in shorter intervals of time than the descending branch. 9. When a pendulum is drawn from its vertical position, it will be accelerated in the direction of the tangent of the curve it would describe, by a force which is as the sine of its angular distance from the vertical position. Thus, the accelerating force at A (fig. 188), would be to the accelerating force at B, as A F to BE. (See art. 5, on the Center of Gravity,) This admits of an easy experimental proof. 10. If the same pendulous body descend through different arcs, iu velocity at the lowest point will be proportional to the chords of the whole arcs described. Thus, the velocity at D, after passing through ABD, will be to the velocity at D after descending through ^e portion BD only, as AD to BD. 1 1 . Farther, the velocity after describing ABD, is to the velocity after describing BD, as v^FD is to v^ED. If, therefore, we would impart to a body a given velocity V, we have only to compute the V* V* height FD, such that FD = — = — r- feet, and through the point F draw the horizontal line FA; then, letting the body descend as a pendulum through the arc ABD, when it arrives at D it will have acquired the required velocity. This property is extremely useful in experiments on the coUision of bodies. CHAP. III.] PENDULUMS. 223 12. The oscillations of penclulums in any arcs of a cycloid are ifockrofud^ or performed in equal times. 13. Oscillations in »n)aU portions of a circular arc are nearly iiockronal, 14. The numbers of oscillations of two different pendulums, in the same time, and at the same place, are in the inverse ratio of the square roots of the length of those pendulums. 15. If / be the length of a single pendulum, or the distance from the point of suspension to the center of oscillation in a compound pendulum, g = the measure of the force of gravity (32^ feet, or 386 inches at the level of St. Paul's* in the latitude of London), / the time of one oscillation in an indefinitely small circular arc, And w =s 3*141593 : then =-v-: <■■' 16. Conformably with this we have 39 \ inches, length of the second \ pendulum 9Jf inches half second in the 4^ inches third of second latitude 2-^^ inches quarter second) of London. 17. Putting y, for the force of gravity in any latitude and at any ^titude, we have also /= -20264 x |^| and 1^1 = 4-9348/ (IL) In other words, whatever be the force of gravity, the length of a second pendnlum, and the space descended freely by a falling body in 1 second, are in a constant ratio, 18. If /, be the length of a pendulum, ^, the force of gravity, and /, the time of oscillation at any other place, then '• ■■■■ ^/i ■■ Vf. '■" ) If the force of gravity be the same, t :t,:: ^l: ^ I, (IV.) If the aune pendulum be actuated by different gravitating forces, we have * At the level of ike mo, in the latitude of liondon, g is 386-289 inches, and the c o rre ^ w m dipg length of the necond penduhim is 39*1393 inches, according to the determination of Major Kater. Conformably with this result are the Biunbers in the Table (in the Appendix), computed at the expense of Messrs. BrawnA and DoniUn, and obli^nngly communicated by them for this work. It has been suspected by M. Beuel'^ and demonstrated by Mr. Francis BaUy^ that, in the refined computations relative to the pendulum, the formulae for the redaction to a vacuom are inaccurate, and that, in consequence, we do not yet pneit^ know the length of a second pendulum. See Phil. Transac. 1832. 224 PENDULUMS. [part II. When pendulums oscillate in equal times in different places, we have gig, :: I : l^. For the variations of gravity in different latitudes, see formulie (XVII., XVIII., XIX.) in the preceding section. 18. If the arcs are not indefinitely short, let v denote the versed sine of the semi-arc of vibration ; then t = , y^i(l + i r + ^j 0- + &c) .... (VI.) In which, when the serai-arc of vibration does not exceed 4 or 5 degrees, the third term of the series may be omitted. If the time of an oscillation in an indefinitely small arc be 1 second, the augmentation of the time will be for a serai-arc of 30° 001675 of 15^* 000426 of 10° 0-00190 of 5° 0-00012 of 2J° 000003 So that for oscillations of 2^° on each side of the vertical, the augmentation would not occasion more than 2^' difference in a day. 19. If D denote the degrees in the semi-arc of au oscillating pendulum, the time lost in each second by vibrating in a circle instead of the cycloid, is ; and consequently the time lost in a whole day of 24 hours, or 24 x 60 X 60 seconds, is ^ D^ nearly. In like manner, the seconds lost per day by vibrating in the arc of A degrees, is J A^. Therefore, if the pendulum keep true time in one of these arcs, the seconds lost or gained per day, by vibrating in the other, will be 4 (I>~ — A^). So, for example, if a pendulum measure true time m an arc of 3 degrees, it will lose 11 1 seconds a day by vibrating 4 degrees ; and 26| seconds a day by vibrating 5 degrees : and so on. 20. If a clock keep true time very nearly, the variation in the length of the pendulum nccessaiy to correct the error will be equal to twice the product of the length of the pendulum, and the error in time divided by the time of observation in which that error is accumulated. If the pendulum be one that should beat seconds, and ^, the daily variation be given in minutes, and n be the number of threads in an inch of the screw which raises and depresses the bob of the pendulum, then X = =■ — ^- i = '05434 n ^ = X « 'i » nearly, for the 24 X 60 ^^ ^ number of threads which the bob must be raised or lowered, to make the pendulum vibrate truly. 21. For civil and military engineers, and other practical men, it is CHAP. III.] CENTEB OF OSCILLATION. 225 highly useful to have Vk portable pendidum^ made of painted tape with a brass bob at the end, so that the whole, except the bob, may be rolled up within a box, which may be enclosed in a shagreen case. The tape is marked 200, 190, 180, 170, 160, &c., 80, 75, 70, Qli, OO, at points, which being assumed respectively as points of suspen- Qon, the pendulum will make 200, 190, &c., down to 60 vibrations in a minute. Such a portable pendulum may be readily employed in experiments relative to falling bodies, the velocity of sound, &c. 22. If the momentum of inertia (§ .5, page 221) of a pendu- lum, whether simple or compound, be divided by the product of the pendulum's weight or mass into the distance of its center of gravity from the point of suspension for axis of motion), the quotient will express the distance of the center of oscillation from the same point (or axis). 23. Whatever the number of separate masses or bodies which constitute a pendulum, it may be considered as a single pendulum, Dvhose center of gravity is at the distance d from the axis of suspen- sion, or of rotation : then, if K' denote the momentum of inertia of tfmt body divided by its mass, the distance O from the axis of rotation to the center of oscillation, or the length of an equivalent mraple pendolnm, will be = £±^.. (VII.) a 24. To find the distanoe of the center of oscillation from the point ^^w axis of suspension, experimentally. Count the number, n, of ^^ecillations of the body in a very short arc in a minute ; then O^l^ (VIII.) Tims, if a body so oscillating made 50 vibrations in a minute; ^'^O^-j^^SS'S^ inches. Or, O = 89 j^ fy in inches, / being the time of one oscillation in a ^ry small arc. If tlie arc be of finite appreciable magnitude, the time of oscilki- ^n must be reduced in the ratio of 8 + versm of semi-arc to 8, 'ore the rule is applied. 25. From the foregoing principles are derived the following ex- sessions for the distances of the centers of oscillation for the several suspended by their vertices and vibrating flatwise, vis. :-— (1.^ Bi^t line or very thin cvlinder, O = | of its length. (2.) IsMceles triangle, O 3= | of its altitude. (3.) Circle, O == | radius. (4.) Common parabola, = 4^^^ altitude. * For tome curiout and vahiabla theorems, by Professor ^try, for the re- ^taecioo oi viknitSani ia the air to those in a VAoiuiin, tee Mr. F. Bailya peper *>rerTed to in the preoeding note. 22i} CKNTKR OF USCILLATION. [PAKT II. (5.) Any parabola, O = X its altitude. 3 m + 1 Bodies vibrating laterally or sideways, or in their own plane : (6.) In a circle, O = } of diameter. (7.) In a rectangle suspended by one angle, O = | of diagonal. (8.) Parabola suspended by its vertex, O = 4 a^tis + J parameter. (9.) Parabola suspended by middle of its base, O = ^ ^^^^ ~^ i parameter. ,,^ N T r . 1 3arc X rad (10.) In a sector of a circle, O = = — = — . ^ 4 chord /,, X T ^ - • (ra<l of base)* (11.) In a cone, O == 4 axis 4- -^ ; — . ^ -^ 5 axis 2 rad^ (12.) In a sphere, O = rad -f rf -f ---= j: ; where d is the 5 (a + rad) length of the thread by which it is suspended. (13.) If the weight of the thread is to be taken into the account, we have the following distance between the center of the ball and that of oscillation, where B is the weight of the ball, d the distance between the point of suspension and its center, r the radius of the ball, w the weight of the thread or ^ire, and 1 the distance of the center of oscillation from the center of gravity, we have ^^ aw^^B)4r''^}w(2dr±d^ (|«j-fB)</ — r» ^ "^ Or, if B be expressed in terms of w considered as a unit, then ^ = ^^ (X.) (14.) If two weights W, W,, be fixed at the two extremities of a rod of given length, S being the center of motion between W and W, ; then, if d equal the distance of the weight W from S, D equal J the distance of W, from S, and m the weight of a unit in length of ^ the rod, we shall have ^ ■" wD« + 2W,D — m</- — 2W</ ^ *^ the radii of the balls being supposed very small in comparison witbrJ the length of the rod. (15.) In the bob of a clock pendulum, supposing it two equaT^ spheric segments joined at their bases, if the radii of those bases b^^ each = ^, the height of each segment c, and d the distance from th^ j point of suspension to the center of the bob, then is '=w- ''^'AV/" <^"-) which shows the distance of the center of oscillation below the ccnteas'J of the bob. CHAP. III.] CENTER OF OSCILLATION. 227 If r the radius of the sphere he known, the latter expression becomes '- "t:^^" <™'-' (16.) Let the length of a rectangle be denoted by /, its breadth by 2 Wy the distance (along the middle of the rectangle) from one end to the point of suspension by ^ then the distance O, from the point of mspensioD to the center of oscillation, ^i-ill be whether the 6gare be a mere geometrical rectangle, or a prismatic metallic plate of uniform density. It follows from this theorem, that a plate of 1 foot long and | of a foot broad, suspended at a fourth of % foot from either end, would vibrate as a half second pendulum. AIMS tliat a plate a foot long, -^^ of a foot wide, and suspended at 1^ of m foot from the middle, would vibrate 30,469 times in 5 hours. Jknd henoBy ike length of a foot may he determined experimentcUly hif vibratume. (17.) If a thin rod» say of a foot in length, have & ball of an inch diameter at each end, A and B, &nd a moveable point of suspension, S; then the tame of oteiUation of such a pendulum may be wnade as long as we please ^ by bringing the point of suspension nearer to the middle of the rod. Or, if tbe point of suspension be fixed, the dis- tance O (and consequently the time of oscillations '^'phicb is as >/0) may be varied by placing A vi«arer to or farther from 8. And this is the prin- ciple of the Metronome^ by wliich musicians some- times regulate their time. (18.) If the weight of the connecting rod be evanescent witli ^e^ard to the weight of the balls A and B ; then if R equal the radius ^f the larger ball, r that of the smaller, D and d the distances of their ■■'oipcctivc centers from 8 : we shall have ^__ B«(5D» + 2R^)-hrM5(/''»H-2r^) ^ 5(DR»)-.rfr') ^^^'^ When R and r are equal, this becomes O = (D + rf) + ^ . ^^ (XVI.) (19.) If the minor and major axes of an ellipse (or of an elliptical P«e of wood or metal) be as 1 to >/ 3, or as 1000 to 1732 ; then, Q 2 228 COMPENSATION TRNDULUMS. [PART IF. if it be suspended at one extremity of the minor axis, the center of oscillation will be at the other extremity of that axis, or its oscilla- tions will be performed in the same time as those of a simple pendu- lum whose length is eqnal to the minor axis. The fnime ellipse also possesses this curious and useful, property^ Tiz. :— That any segment or any £one of the ellipse cut oflf by lines parallel to the major axis, whether it be taken near the upper part of the minor axis, near the middle, or near the bottom of the same, will vibrate in the same time as the whole ellipse^ the point of sus- pension being at an extremity of the minor axis. 26. It is evident from § 1 7, page 228, that pendulums in differ- ent latitudes require to be of different lengths, in order that they may perform their vibrations in the same time; but besides this there is another irregularity in the motion of a pendulum in the same place, arising from the different degrees of temperature. Heat ex- panding, and cold contracting the rod of the pendulum, certain small variations must necessarily follow in the time of its vibration ; to remedy which, Tarious methods have been invented for construct- ing what are commonly called compensation pendulums^ or atich as shall always preserve the same distance between the center of oscil- lation and the point of suspension ; and of these we shall describe two or three. Comf^ensalion pendulums have received different denominations, from their form and materials, as the gridiron pendulum^ mercurial "^ pendulum, &c. 27. The gridiron pendulum consists of five rods of steel, and four -m- ,i of brass, placed in an alternate order, the middle rod being of steel, ..^ f J, by which the pendulum ball is suspended ; these rods of brass andf^ d steel are placed in an alternate order, and so connected with eachn^-h other at their ends, that while the expansion of the steel rods has s^ a tendency to lengthen the pendulum, the expansion of the brass rods .Ezds acting upwards tends to shorten it. And thus, when the lengths o ^n^oi the brass and steel rods arc duly proportioned, theur expansions an<» mznd contractions will exactly balance and correct each other, and so pre^^»-e- serve the pendulum invariably of the same length. Sometimes 3, t w^~ 7, or 9 foda, are employed in the construction of the gridiron penduKi^v .u- lum ; and zinc, silver, and other metals may be used instead of bras=%.^BS8 and steel. 28. The mercurial pendulum was invented by Mr. Graham, dt^w^ an eminent clockmaker, about the year 1715. Its rod was made ^ of brass, and branched towards its lower end, so as to embrace a cyliB^ci Ji- dric glass vessel 13 or 14 inches long, and about 2 indiea diamet er- ^^r ; which, being filled about 12 inches deep with mercury, forms iJ^B ie weight or ball of the pendulum. The height of the mercury in ti^We glass being so proportioned to the length of the rod that its expansi^*^^^ and contraction exactly balanced the expansion and contraction ^^1^ the pendulum rod, and preserved the distance of the center of osc ^■'Z' lation from the point of suspension invariably the same. This kind of pendulum fell entirely into disuse soon after Graham's / CHAP. III.] GYBATION AND ROTATION. 229 time ; but it has lately been reniclopted with considerable success by practical astronomers. A very instractiTe paper on its principles, construction, and use, has been published by Mr. F, Bailjfy in vol. i. part 2, Memoirs of the Astronomical Society of London; in which paper is also contained an extensive and valuable table of the expan- sion of different substances by beat. 29. Reid's compensation pendulum is a recent invention of Mr. Adam Beid, of Woolwich, the construction of which is as follows : — A N (fig. 189) is a rod of wire, and Z Z a hollow tnbe of zinc, which slips ou the wire, being stopped from falling off by a nut N, on which it rests ; and on the upper part of this cylinder of zinc rests the heavy ball B: now the length of the tube ZZ being so adjusted to the length of the rod A N, that the expansions of the two bodies shall be equal with equal degrees of temperature; that is, by making the length of the zinc tube to that of the wire, as the expansion of tlie wire is to tliat of zinc, it is obvious that the ball B will in all cases pre- serve the same distance from A; for just so much as it would descend by the expansion of the wire downwards, so much will it ascend by the expansion of the zinc upwards, and consequently its vibrations will in all temperatures be equal in equal times. 30. Drummond s compensation pendulum was proposed by an artist of that name, in Lancashire, more than 70 years ago. A bar of the same metal with the rod of the pendulum, and of the same thick- ness and length, is placed against the back part of the clock case ; from the top of this a piece projects, to which the upper part of the pendulum is connected by two fine pliable chains or silken strings, which just below pass between two plates of brass whose lower edges will alwavs terminate the lengtli of the pendulum at the upper end. These plates are supported on a foot fixed to the back of the case. This bar rests upon an immoveable base on the lower part of the case, and is braced into a proper groove, which admits of no motion any way but that of expansion and contraction in length by heat and cold. In this construction, since the two bars are of equal magnitude and of the same material, their expansions and con trac- tions will always be eqnal and in opposite directions ; so that one will serve to correct and annihilate the effects of the other. CENTER OP OYBATION AND THE PRINCIPLES OP ROTATION. 1. The center of gyration is that point in which, if all the matter contained in a revolving system were collected, the same angular velocity would be generated in the same time by a given force acting at any plaee as would be generated by the same force acting similarly ID tbe body or system itself. When the axis of motion passes through the center of gravity, tlien is the center called the principal center of gyration. 2. The distance of the center of gyration from the point of sus- pensioD or the axis of motion, is a mean proportional between the distances of the centers of osdllation and gravity from the same point or axis. r" 230 GYRATION AND ROTATION. [PART 11. If 6 equal the distance of tbe center of gravity from the point of suspension, O the distance of tbe center of oscillation, and R the distance of the center of g\Tation from the same point, then we have R = >/go (XVII.) 3. The distance R of tlie center of gyration, from the center or axis of motion, in some of the most useful cases, is given below. In a circular wheel of uniform thickness ... R = rad >/ \, 111 the periphery of a circle revolving about) _^ A / X the diameter j v s* In the plane of a circle ditto R = | rad. In the surface of a sphere ditto R =rad s/ §• In a solid sphere ditto R = rad ^/t =^ i^rnearly. In a plane ring formed of circles whose radii) /** "*" are R, r, revolving about its center j ^\/ 2~ In a cone revolving about its vertex R = J ^ 3* ^' "+" J »''• In a cone revolving about its axis R = rv^-j^y. In a paraboloid R = r ^^ i- / R' + f^ In a straight lever whose arms are R and r, R = a / — r ; • ^ . V srR + r) 4. If the matter in any gyrating body were actually to be placed as if in the center of gyration, it ought either to be disposed in the circumference of a circle whose radius is R, or at two points R, R^ diametrically opposite, and each at the distance R from the center. 5. By means of the theory of the center of gyration, and the values of R = ^, thence deduced, the phenomena of rotation on a fixed axis ^ become connected with those of accelerating forces : for then, if a ,^ weight or other moving ])ower P act at a radius r to give rotation to ^y a body whose weight equals W, and the distance of whose center of "^ gyration from the axis of motion equals ^, we shall have for the -^^-^ accelerating force, the expression /=P^^ (^v"^> and consequently for the space described by the actuating weight or "tk-^di power P, in a given time ty we shall have the usual formulse (V.) and f» jd (VII.), page 214, introducing the above value of/. 6. In the more complex cases, the distance of the center of gyra -^- tion from the axis of motion may best be computed from an experi ^" ment. Let motion be given to the system, turning upon a horiatontal^-^ axis, by a Mcight P acting by a cord over a pulley or wheel whose^^^ radius equals r, fixed upon the same axis, and let s be the space ^^^ through which the weight P descends in the time /, the proposed ^^ body whose weight is W turning upon the same axis with the wune— ^ angular velocity ; then i CHAP. III.] AXES OP ROTATION. 231 K=,.^€£f!lz^ (--) Example. A body which weighs 100 lbs. turns upon a horizontal axis, motion being communicated to it by a weight of 10 lbs. hanging from a very light wheel of 1 foot diameter. The weight descends 2 feet in 3 seconds. Required the distance of the center or circle of gyration from the axis of motion. " Here, potting y = 32, instead of 32^, we obtain as an approxima- tiye result, R = ^ /32xl0x9xi-4xl0xi ^ ^.33^3 ,^^^ V 4 X 100 I. When the impulse communicated to a body is in a line passing through its center of gravity, all the points of the body move forward with the same velocity, and in lines parallel to the direction of the impulse communicated. But when the direction of that impulse docs not pass through the center of gravity, the body acquires a rotation OD an axis, and also a progressive motion, by which its center of gravity is carried forward in the same straight line, and with the same velocity, as if the direction of the impdse had passed through the center of gravity. The progressive and rotatory motion are independent of one another, each being the same as if the other had no existence. 8. When a body revolves on an axis, and a force is impressed, tending to make it revolve on some other, it will not revolve on either, but on a line in the same plane with them, dividing the angle which they contain, so that the sines of the parts are in the inverse ratio of the angular velocities with which the body would have revolved about the said axes separately. 9. A body may begin to revolve on any line as an axis that passes through its center of gravity, but it will not continue to revolve per- manently about that axis, unless the opposite rotatory forces exactly balance one another. This admits of a simple experimental illustration. Suspend a thin circular plate of wood or metal by a cord tied to its edge, from a hook to which a rapid rotation can be given. The plate will at first turn upon an axis which is in the continuation of the cord of rotation ; but as the velocity augments, the plane will soon quit that axis, and revolve permanently upon a vertical axis passing through its center of gravity, itself having assumed a horizontal position. The same will happen if a ring be suspended, and receive rotation in like manner. And if a flexible chain of small links be united at its two ends, tied to a cord and receive rotation, it will soon adjust itself so as to form a ring, and spin round in a horizontal plane. Also, if a flattened spheroid be suspended from any point, how- ever remote from its minor axis, and have a rapid rotation •:ivcn it, it nill ultimately turn upon its shorter axis ])08itod vcrticully. This evidently serves to confirm the motion of the earth upon its shorter axis. 232 CENTHAL FORCES. [PABT II 10. In every body, however irregular, tbere are three axes of per mancnt rotation, at right angles to one another. These are callei the principal axes of rotation; and they have this remarkable pro- perty, that the raomentam of inertia with regard to any of them i either a maximum or a minimum. CENTRAL FORCES. 1 . Centripetal force is a force which tends constantly to solicit o to impel a body towards a certain fixed point or center. 2. Centrifugal force is that by which it would recede from such center, were it not prevented by the centripetal force. 3. These two forces are, jointly, called central forces. 4. If W denote the weight of a body moving in a circle* whoa radius equals r, with the velocity t?, its centrifugal force scy will hm f'"-^ ■ ■ ■■- (-•) where g equals the force of gravity. If t equal the time of one revolution, and ir = 31 41 59, then /=i^=-!^w («,., 5. When a body describes a eirde by means of a force directed its center, its actual velocity is every where equal to that which, would acquire in falling by the same uniform force through half fc radius. 6. This velocity is the same as that which a second body woi^ acquire by falling through half the radius, whilst the first described portion of the circumfereuce equal to the whole radius. 7. In equal circles the forces are as the squares of the times » versely. 8. If the times are equal, the velocities are as the radii, and C forces are also as the radii. 9. In general, the. forces nre as the distances or radii of the drcf directly, and the squares of the times inversely. 1 0. The squares of the times are as the distances directly, and tl forces inversely. 11. Hence, if the forces are inversely as the squares of the di tances, the squares of the times are as the cubes of the distance That is, if F :/ :: rf« : D«, then T- : f ;: D' : d\ . . (XXII.) 12. The right line that joins a revolving body and its center of a traction, called the radius vector^ always describes eqnal arcaa in eqa times, and the velocity of the body is inversely as the perpendical drawn from the center of attraction to the tangent of the t^nrve the place of the revolving body. 13. If a body revolve in an elliptic orbit by a force directed to o of the foci, the force is inversely as the square of the distance: « CHAP. III.] CENTRAL FORCES. 233 the mean distances and the periodic times have the same relation as in Art. 11. This eomprtkenda the case of the planetary motions. 14. If the force which retains a hody in a curve increase in tlie simple ratio as the distance increases, the body will still describe an ellipse; but the force will in this case be directed to the center of the ellipse; and the body in each revolution will twice approach towards it, and again twice recede from that point. 15. On the principles of central forces depends the operation of the conical pendulum applied as a governor or regulator to steam engines, water mills, &c. This contrivance will be readily comprehended from fig. 1 90, where A a is a vertical shaft capable of turning freely upon the sole a. C D, e F, are two bars which move freely upon the center C, and carry at their lower extremities two equal weights, P, Q; the bars CD, C F, are united, by a proper articulation, to the bars O, H, which latter arc attached to a nng 1, capable of sliding up and down the vertical shaft A a. When this shaft and connected apparatus are made to revolve, in virtue of the centrifugal force, the balls P Q fly out more and more from A a, as the rotatory velocity increases: if, on the con- trary, the rotatory velocity slackens, the balls descend and approach A a. The ring I ascends in the former case, descends in the latter: and a lever connected with I may be made to reguhUe the energy of the moving power. Thus, in the steam engine, the ring may be made to act on the valve by which the steam is admitted into the cylinder; to augment its opening when the motion is slackening, and recipro- cally diminish it when the motion is accelerated. The construction is often so modified, that the fiying out of the balls causes the ring I to be depressed,, and vice versd; but the gene- ral principle is the same. Here, if the vertical distance of P or Q below C, be denoted by c/, the time of one rotation of the regulator by ^, and 3- 14 1593 by ^, the theory of central forces gives / = 2 ^ /v/s^ = ^'^^^^^ >/ rf . . (XXIII.) Hence, the periodic time varies as the square root of the altitude of the conic pendulum, let the radius of the base be what it may. Also, when ICQ = ICP=s 45°, the centrifugal force of each ball is equal to its weight. 16. As the practical utility of the conical pendulum depends in a great degree upon iu sensibility, or the change which must take place m its Telocity before it will move the ring 1 to the required extent, we subjoin the following formula for determining the weight of the balls P and Q, for any degree of sensibility which may be required. LetW equal the weight of both the balls, P the power required to move the valve (or produce whatever effect may be required) when applied to the ring in the direction of the spindle, J^, the number of revolutions which the governor is intended to make per second, N^ the number of revolutions which the same must make to move the 234 iNQriitiEs cuNNfccTiiii WITH [part II. ring I, </ as before the vertical distance from the plane of the balls P and Q to the point C, b tlie distance CD or C F, / the distance C P, a the distance of the point D from the axis of the spindle, c the vertical distance from the plane joining the points D and F from the point I, and r the radius of the circle described by the balls, or their distance from the axis of the spindle ; then '-iJ^is? ^^^'^•> If the distance C D is made equal to D I, so that the four rods form a ])aralleIogram, this last formula becomes ^^' = i-.-.73r</?(N?-3]^) • • • (^^^'-^ 17. In the foregoing formulte the center of oscillation of the balls and system of rods is assumed to coincide with the center of the balls, an assumption not strictly correct, although sufficiently so for all practical purposes. Should, however, greater accuracy be required, the true center of oscillation having been found, the following sub- stitutions must be made, when the preceding formulae will give an ac- curate result ; viz., for /, the distance from C to the center of oscilla- tion, and for r the distance of the center of oscillation from the ver- tical axis of the spindle. INQUIRIES CONNECTED WITH ROTATION AND CENTRAL FORCES. 1. Suppose the diameter of a grindstone to be 44 inches, and iisa weight half a ton ; suppose also that it makes 386 revolutions in tm minute. What will be the centrifugal force, or its tendency to burst ^ 44 386 Here the velocity = 75 X 31416 x --;--- = 74106; then by substituting these values in formula (XX.) we have ^ 74-1062 X -5 /= - 7 -- = 46-8 tons. •^ 32 X -tj the measure of the required tendency. 2. If a fly wheel 12 feet diameter, and 3 tons in weight, revolivi in 8 seconds : and another of the same weight revolves in 6 seconds what must be the diameter of the last, when their centrifugal force the same ? By formula (XXI.) F : / : : ~ : ^. Therefore, since F is = ,,^ D d , Vt' 12 X 36 ^, ^ 2^ = ;;}> ^»' « = -^P" = J54— = ^J ^^^^ ^^^ answer. CHAP. III.] ROTATION AND CENTRAL FORCES. 235 3. If a fly of 12 feet diameter revolve in 8 seconds, and another of the same diameter in 6 seconds : what is the ratio of their weights when their central forces are equal ? By § 7, page 232, the forces arc as the squares of the times in- versely when the weights are equal : therefore, when the weights are unequal, tbey must be directly as the squares of the times, that the central forces may be equal. Hence » : W :: 36 : 64 :: 1 : 1^ That is, the weight of the more rapidly to that of the more slowly revolving fly, must be as 1 to 1^, in the case proposed. 4. If a fly 2 tons weight and 1 G feet diameter, is suflicient to regu- late an engine when it revolves in 4 seconds; what must be the weight of another fly of 12 feet diameter revolving in 2 seconds, so that it may have the same power upon the engine ? Hertf, by § 9, page 232, we must have — — = -^ ; therefore 40cwt. X 16 X 4 160 ,^, 1^ . , s - - ^ ^ = -;« = 134 cwt., the weight 12 X 16 12 3 » 6 of tbe smaller fly. NoUj'-^K fly should always be made to move rapidly. If it be intended for a mere Regulator, it should be near the fimt mover. If it be intended to accumulate force in tbe workittg pointy it must not be far separated from it. 5. Given the radius R of a wheel, and the radius r of its axle, the weigbt of both, tp, and the distance of the center of gyration from the axis of motion, ^ ; also a given power P acting at the circumference of the wheel ; to find the weight W raised by a cord folding about tbe axle, so that its momentum shall be a maximum. Here ^_ s/ (R^ P* -f 2 R*Pg* w - h e^y* - f P»Rrg* -f P*RV)— R'P— g*ig Cor. 1. When R =s r, as in the case of the single fixed pulley, (ben W= >/(2P»R' + 2RPf*»+^»« -h P»Re») — ^»— P. R R" Cor. 2. When the pulley is a cylinder of uniform matter e* = J R% ^•^d the expression becomes W = VlR' (2 P« -h 4 P » + 4 tP*)} — i w — P. 6. Let a given power P be applied to the circumference of a wheel, ^'liose radius equals R, to raise a weight W at its axle, whose radius ^^ r, it is required to find the ratio of R and r when W is raised with the ^^^eatest momentum ; the characters W and ^ denoting the same as in ^^^c last proposition. 236 COLLISION OF BODIES. [PART I Here r = ; — r • P (y -r W) Cor. Wlien the inertia of the machine is eTanesoent, with retp© to that of P -I- W, then is r = R ^ / (\ + ^) — 1- 7. In any machine whose motion accelerates, the weight ^-ill I moved with the greatest velocity when the velocity of the power is 1 that of the weight as 1 + P a / (^ + T^ ) ia to 1 ; the inertia i the machine heing disregarded. 8. If in any machine whose motion accelerates, the deseent of oc weight causes another to ascend, and the descending weight be give- the operation being supposed continually repeated, the effect vrill 1 greatest in a given time when the ascending weight is to the du scending weight, as 1 to 1*618, in the case of equal heights; and other cases when it is to the exact counterpoise in a ratio which^ always between I to 1| and 1 to 2. 9. The following general proposition with regard to rotatory nz lion will be of use in the more recondite cases. If a system of bodies be connected together and supported at «■ point which is not the center of gravity, and then left to descend that part of their weight which is not supported, 2g multiplied \m the sum of all the products of each body into the space it has p^- pendicularly descended, will be equal to the sum of all the prodiB of each body into the square of its velocity. Sect. IV. Percussion or Collision of Bodies in motion. 1. In the ordinary theory of percussion, or collision, bodies - regarded as either hard^ sojly or elastic, A hard body is that wh— parts do not yield to any stroke or percussion, but retains its fig* unaltered. A soft body is that whose parts yield to any stroke impression, without restoring themselves again, the shape of the be: remaining altered. An elastic body is that whose parts yield to m stroke, but presently restore themselves again, so that the body gains the same figure as before the stroke. When bodies which he been subjected to a stroke or pressure return only in part to i\m original form, the elasticity is then imperfect: but if they rest* themselves entirely to their primitive shape, and employ just as mu. time in the restoration as was occupied m the compression, then the elasticity perfect. It has been customary to treat only of the collision of bodies p^ fectly hard or perfectly elastic : but as there do not exint in nat* any bodies (which we know) of either the one or the other of th kinds, the usual theories are but of little service in practical i* CHAP. III.] COLLISION OP B0DTK8. 237 cbanics, except as tliey may suggest an extension to tbe actual cir- cumstances of nature and art. 2. Tbe general principle for determining tbe motions of bodies from percussion, and wbicb belongs equally to both elastic and non-elastic bodies, is this : viz. that there exists in the bodies the same momentum, estimated in any one and the same direction, both before the stroke and after it. And this principle is the immediate result of the law of nature or motion, that reaction is equal to ac- tion, and in a contrary direction ; from whence it happens, that what- ever motion is communicated to one body by the action of another, exactly the same motion does this latter lose in the same direction, or exactly the same does the former communicate to the latter in tlie contrary direction. From this general principle too it results, that no alteration takes place in the common center of gravity of bodies by their actions upon one another ; but that the said common center of gravity per- severes in the same state, whether of rest or of uniform motion, both before and after the impact. 3. If the impact of two perfectly hard bodies be direct, they will, after impact, ekher remain at rest, or move on uniformly together with different velocities, according to the circumstances under which they met. Let B and h represent two perfectly hard bodies, and let the velo- city of B be represented by V, and that of b by r, which may be taken either positive or negative, according as h moves in the same direction as B, or contrary to that direction, and it will be zero when b is at rest. This notation being understood, all the circumstances of the motions of the two bodies, after collision, will be expressed by the formula : , . BV±bv velocity = — B -f- irliich being acoommodated to the three circumstances under which 9 may enter, become „ V , , BV -f 5t? / when both bodies moved in (I.) velocity = p^^ \ the same direction ,,^ . , . BV — bv i when the bodies moved in (II.) velocity = p^^ [ contmry directions ,^ ^ , . B 4- ft ( when the body b was at (HI.) velocity =:^^ I rest. Tbese formulsB arise fi^m the supposition of the bodies being per- fecOj hard, and consequently that the two after impact move on uni- formly together as one mass. In cases of perfectly elastic bodies, other foraulflB have place which express the motion of each body Mparately; as in the following proposition. 4. If the impact of two perfectly elastic bodies be direct, theii- relative velocities will be the same both before and aft^r impact, or 238 COLLISION OF BODIES. [pAHT they will recede from eau^ other with the same velocity with whi- they met; that is, they will be equally distant, in equal times, bo^ before and after their collision, althou^ the absolute velocity of etu^^ may be changed. The circumstances attending this change of m.^ tion in the two bodies, using the above notation, are expr^eed in t~l^ two following formulae : ^~X — ^— = velocity of B . . . (IV.) 2BV + (B-6)v , . ^, ^ ^— = velocity of 6 . . . (V.) which needs no modification, when the motion of & is in the s^^ direction with that of B. 5. In the other case of b's motion, the general formulae become^- ^^A!Ll_(Ezl*)^ = ,elocity«fB. (VI.) 2_LL^J!L:ii)^=velockyofft . (VII.) when b moves in a contrary direction to that of B, which arises fr€>i taking v negative. And (B — b)y ^ -— = velocity of B ( VIII.) n n ^ - - - = velocity of ^ (IX.) when b was at rest before impact, that is, when v = 0. G. If a perfectly hard body B, (fig. 191,) impinge obliquely uf?^ a perfectly hard and immoveable plane A D, it will after coUi^'^ move along the plane in the direction C A. And its velocity before impact Is to its velocity after impact As radius Is to the cosine of the angle BCD. But if the body be elastic it will rebound from the plane in t direction C £, with the same velocity, and at the same angle wi - which it met it, that is, the angle ACE will be equal to the an^ BCD. 7. The force with which a body impinging obliquely strikes plane, is to the same if it had acted perpendicularly, as the sine ^ the angle (B C D) of incidence, is to radius. 8. In the case of direct impact, if B be the striking body, b tH body struck, V and v their respective velocities before impact, V, ax^ «, their velocities afterwards; then the two following are genei^ formulce: viz. CHAP. III.] ON THK MECHANICAL lOWFRS. 23.9 ' v- r, = r -f n In these, if « = 1, they serve for non-elastic bodies ; if « = 2, for bodies perfectly elastic. If the bodies be imperfectly elastic, n has some intermediate Talue. When the body struck* is at rest, the preceding equations become V. = V - -!Lll (XII.) » B+b ^ ' r,=!^ (XIII.) n = ^._(« +i) (XIV.) BV ^ ' from which the value of n may be determined experimentally. 9. In the usual apparatus for experiments on Collision, balls of different sizes and of various substances are hung from different points of suspension on a horizontal bar. MAN (fig. 192) is an arc of a circle whose center is S ; and its graduations, 1, 2,3, 4, 5, &c., indicate the lengths of chord lines measured from Uie lowest point D. Any ball, therefore, as P, may be drawn from the vertical, and made to strike another ball hanging at the lowest point, with any assigned velocities, the height to which the ball struck ascends on the side A M furnishing a measure of its velocity ; and from that the value of n may be found from the last equation. Balls not required in an in- diridoal experiment, may be put behind the frame as shown at A and B. The cup C may be attached to a cord, and carry a ball of clay, &c. when required. Example, Suppose that a ball weighing 4 ounces strikes another ball of the same substance weighing 3 ounces, with a velocity of ] 0, ^d communicates to it a velocity of 8| : what, in that case, will be ^he value of n ? «eren= ' p^ = ^-^^^ = ^j^- = 144375 the index of the degree of elasticity; perfect elasticity being indicated by 2.. Sect. V. On the Mechanical Powers, 1. The most complicated machinery is nothing more than a com- bination, or constant repetition, of a few simple mechanical expe- dients for modifying and changing the direction of the several forces 240 MECHANICAL POWERS. [PART II. or pressures, transmitted by them through the machine. A certain pressure or force, being communicated to what is termed the first or prime mover of the machinery, is thus transmitted through every part of the machinery, being regulated and modified in such a man- ner as to produce the effect required, at what is termed the reork- ing point. Thus, by means of the machine termed a crane, a man, by applying the pressure of his hand (amounting to perhaps about 30lb6.) to the handle of the crane, is enabled, through the intervention of the machinery, to raise an enormous weight, as, for instance, say 12,000 lbs., which, without some such expedient, would defy all his efforts to move. 2. These elementary parts of which more complicated machinery is composed, have been called the mechanical powers^ a term which is liable to lead to a misapprehension of the effects which they are really capable of producing. For, ^continuing the use of the previous illustration,) although the man, through the instrumentality of the crane, by exerting a ^rce of SOlbs., is enabled to lift 12,000lbs., or four hundred times the amount, it will be found that his hand will move four hundred times as fast as the weight, and, therefore, what advan- tage he may appear to eain in the weight which he is enabled to •« lift, he really loses again by the length of time which it takes him to ^ raise it to any given height ; and which would be found by experi- — ment to be just what he would require to raise separately four hnn- — dred weights, each of dOlbs., to the same height. 3. The nnmber of the mechanical powers is usually reckoned to ^ be six : viz. the lever ^ the wheel and a^rfe, the pttUey, the indined^^ planey the wedge, and the screw. 4. In treating of these machines, we use the word power^ to denote ^s the force which is supposed to be exerted at the origin of the machine, ^.i^ and the M'ord weight to denote the effect which that force produces^^ at the working point of the machhie : and we shall express them by "^ the letters P and \V respectively. LEVERS. 1 . A lever is an inflexible bar, whether straight or bent, and sup^ posed capable of turning upon a fixed, unyielding point, called aful^ crum. There are three kinds of levers. 2. When i\\t fidcrum is between the power and the weight, as iia fig. 193, the lever is said to be of \\\q first kind, 3. When the weight is between the power and the fulcrum, as in fig. 194, the lever is of the second kind. 4. When the power is between the weight and the fulcrum, as in fig. 1 95, the lever is of the third kind. The bent lever, as employed in the operation of drawing a nail with a hammer, is sometimes considered as a fourth kind, but is really a lever of the first kind. .5. In all these coses, where there is an equilibrium, it is indicated by this general property, that the product of the weight into the dis- tance at which ft acts, is equal to the product of the power into the CBAf.lu,] MECHANICAL POWERS. 241 distance at which it acts : tke distances being estimated in directione P^'yndieidar to those in which the weight and power act respectively. 3T>|», ID each of the three preceding figurcR, P. AF = W. BF, *• the power and weight are reciprocally as the distances at which ^cyact. If, in fig. 193y for example, the arm A F were 4 times F B, 4 Ihs. ftngfng at B would he halanced hy 1 Ih. at A ; and if A F were 5 Enes FB, 1 Ih. at A would halance 5 Ihs. at B; and so on. 6. If several weights hang upon a lever, some on one side of the ilcnun, some on the other, then there will be an equilibrium, when « sum of the products of the weights into their respective distances 1 one ade, is equal to the several products of weights and distances la the other side. Tor, the product of the weight into its distance from the fulcrum \ is the same as its trioment about the point F, and therefore the OTegmng proposition is a direct consequence of the principle of the e«)Yiality of the moments of any forces in equilibrium, about any fixed point, as explained at § 5, page 189. 7. When the weight of the lever is to be taken into the account, F^oceed just as though it were a separate weight suspended at the ^We of its center of gravity. 8. When two, three, or more levers act one upon another in sue • lesion, then the entire mechanical advantage which they afford, is foond by taking, not the eum^ but the product of their separate ad- stages. Thus, if the arms of three levers, acting thus in con- P^on, are asStol, 4to1, and 5 to 1, then the joint advantage » that of 3 X 4 X 5 to 1, or 60 to 1 : so that 1 lb. would, through ^ intervention, balance 60. 9. Id the first kind of lever the pressure upon the fulcrum = P t W; in the other two it is = P '^ W. 10. Upon the foregoine principles depends the nature of scales ■Bd beams for weighing all bodies. For, if the distances be equal, tko will the weights be equal also ; which gives the construction of tlie common scales. And the Roman statera, or steel-yard, is also a Uifer^ but of unequal arms or distances, so contrived that one weight anJy may serve to weigh a great many, by sliding it backwards and Avwsrds to different distances upon the longer arm of the lever. In tfce common halanc-e^ or scales, if the weight of an article when •soertained in one scale is not the same as its weight in the other, Mr square root of the product of those two weights will give the true metgit. ]]. From nnmeroos examples of the power and use of the lever, we which shows its manner of application in the printing-presses of dbe late Earl Stanhope may be advantageously introduced. 242 MECHANICAL POVfBBS. [part II. In the adjoining figure, let A BCD be the general frame of the press, connected bv the cross pieces N O, DC. E is a center connected with the frame by the bars E N, E R, E O. To this center arc fixed a bar KL, and a lever E F, to which the hand is applied when the press is nsed. Tiiere are also several other pieces connected by joints at N, 6, 1, K, L, M, O, H, which are so adjusted to each other, that when the hand is applied to the lever EF at F, by ^\^^ f)ressing it downwards KL is brought into a horizontal line or paral- ^JTg-^ el to O U or D C, in which situation N I O, O M H, also form each omm j straight line. It is evident that the nearer these different pieces, t^"i a^ | above mentioned, are to a straight line the greater is the lever EF. ~=s^ f in proportion to the perpendicular KS at the other end of the lever ^^ -^^ EK, formed by a perpendicular from K falling on F£ prodocedE:^ •?(/. Consequently a small force applied at F will be sufficient to produce ^g very great effect at K, when I K, K E are nearly in a straight line, an .^i^ajif so on, for the other pieces above mentioned. Hence the force applied by hand at F must be very considerable ^/e in forcing down OH, which slides on iron cylindrical bars, or ? in pressing any substance placed in the aperture PQ, between the be- ^^ or plate and the frame DC. This contrivance is now often introduced into mechanism, und er the name of the toggle ^ or knee-Joint, WHEEL AND AXLE. 1. The nature of this machine is suggested by its name. To it may be referred all turning or wheel-machines containing wheels of different radii ; as well-rollers and handles, cranes, capstans, \iir^ ^' lasses, &c. 2. In the wheel and axle the mechanical property is the sam^ u in the lever: viz. P . AC = W . BC (fig 196); and the reaaoim i» evident, because the wheel and axle is only a kind of perpetrnv/ lever. 3. When a series of wheels and axles act upon each other, so ^ to transmit and accumulate a mechanical advantage, whether the communication be by means of cords and belts, or of teeth nod pinions, the weight will be to the power, not as the mm, but as tbe continual product of the radii of the wheels to the continual product of the radii of the axles. Thus, if the radii of the axles, a,ft,e,A<i (fig. 197,) be each three inches, while the radii of tlie whedif A, B, C, D, £, be 8, 6, 9, 10, and 12 inches respectively: tbtn W:P::9xex9xlO x 12 : 3 X 3 X 3 X 3 X 3 :: 240:1. i computation, however, in which the effect of friction is disr^arded. PULLEY. palley is a small wheel, commonly made of wood or brass, of turning upon an iron axis passing through its center, and a block, and the use of which is, that by means of a cord ound its circumference, we are enabled to alter the direction pce in any way that we choose. The pulley is either single purpose, or combined with others to obtain a mechanical ad- . It is also either fixed or moveable, according as it is fixed ilacc, or moves up and down with the weight or power*. a power sustain a weight by means of a fixe<l pulley, as in , the power and weight are equal. lere are several different methods of combining pulleys to- or forming as they are termed separate systems of pulleys, for g 4 mechanical advantage. ^ first system is shown in fig. 199, in which there are two f pulleys, the upper fixed, and the lower attached to the and rising with it ; only one continuous rope is employed, lay be attached to either block, and passed successively round dley. In this system, the weight sustained at W, is to the sustaining it at P, as the number of ropes engaged between blocks, is to 1 ; thus in fig. 199 a, the mechanical advantage, I, and in fig. 199 5, it is 4 to 1. be second system is when there are as many separate ropes ! are pulleys, each rope being attached to a fixed point at one I, passing under one of the pulleys, has its other end attached lock of the next pulley above it. This system is represented •elul eombinaUon of the wheel and axle, a fixed and a moveable pulley ted in the marginal dia- rhe loadf as of stones or >4b » Imild a wall, is raited from F ^^ tkufs a rope BP Lis fixed UL *_ - I U I* I — CHAP. III.] MECHANICAL POWERS. 245 Tkas, suppose the angle A B H was 30°, D B I 6*0% and con* seqiiently ABD 90°: since tlic natural sines of 90% 60°, and 30°, are 1, -866, and -5 respectively, or nearly as 100, 86*6, and 50; if the heavy body weigh 100 lbs., the pressure upon AB would be 86'f> 1bs.i and upon BD 50 lbs. This proposition is of very extensive utility, comprehending the pressure of arches on their piers, of buttresses against walls, or upon the ground, &c., because the circumstance of one of the pianos becoming either horizontal, or vertical, will not affect the general relation above exhibited. WEDOB. 1. A wedge is a triangular prism, or a solid conceived to be generated by the motion of a plane triangle parallel to itself upon a straight line which passes through one of its angular points. The wedge is called isosc^es^ rect- angtUar, or 9calene^ according as the generating triangle IS isosceles, right-angled, or scalene. It is very fre- quently used in cleaving wood, as represented in the figure, and often in raising great weights. 2. When a resistmg body is sustained against the face of a wedge, by a force acting at right angles to its direction ; in the case of equi- Hbrium, the power is to the resistance as the sine of the semi-angle of the wedge, is to the sine of the angle which the direction of the resistance makes with the face of the wedge; and the sustaining force will be as the cosine of the latter angle. 3. When the resistance is made against the face of a wedge by a body which is not sustained, but will adhere to the ])lace to which it is applied without sliding, the power is to the resistance, in the ease of equilibrium, as the cosine of the difference between the semi-angle of the wedge and the angle which the direction of the resistance makes with the face of the wedge, is to radius. 4. When the resisting body is neither sustained nor adheres to the point to which it is applied, but slides freely along the face of the wedge, the jwwer is to the resistance as the product of the sines of the semi-angle of the wedge and the angle in which the resistance ia inclined to its face is to the square of radius. SCREW. 1. The screw is a spiral thread or groove cut round a cylinder, and erery where making the same angle with the length of it. So that if the surface of the cylinder, with this spiral thread on it, were un-» ^Ided or developed into a plane, the spiral thread would form a ^rmight inclined plane, whose length would be to its height^ as the ^'rcnmference of the cylinder is to the distance between two threads ^f the screw : as is evident by considering that, in making one '^und, the spiral rises along the cylinder the distance between the ^^o threads. 2, The energy of a power applied to turn a screw round, is to ^W force witli which it presses upward or downward (setting aside 246 MECHANICAL POWERS. the friction), as the distance between two threads is to th ference where the power is applied : viz., as the circumferei is to the distance BI (fig. 205). 3. The endless screw^ or perpetual serew^ is one which and tnms a toothed wheel DF (fig. 206), without a concave screw; being so called because it may be turned for crei coming to an end. From the diagram it is evident that screw turns once round, the wheel only advances the d one tooth. 4. If the power applied to the lever, or handle of a screw, A B, be to the weight, in a ratio compounded of the of the axis of the wheel, EH, to the periphery describe power in turning the handle, and of the revolutions of the i to the revolutions of the screw CB, the power will bn weight. Hence, 5. As the motion of the wheel is very slow, a small p raise a very great weight by means of an endless screw. A fore the chief use of such a screw is, either where a great to be raised through a little space, or where only a si motion is wanted. For which reason it is very 8er\nceabh and watches. The screw is of admirable use in the mechanism of mi and in the adjustments of astronomical and other instrun refined construction. 6. The mechanical advantage of a compound machini determined by analyzing its parts, finding the mechanical of each part severally, and then blending or compound]: ratios*. Thus, if m to 1, n to 1, r to 1, and e to 1, separate advantages; then mnr s io 1, will measure the ad the system. * The marginal representation of a common construction of a crane to raise heavy loads, will serve to illustrate this. By human energy at the handle a, the pinion h is turned; that gives motion to the wheel W, round whose axle, r, a cord is coiled; which cord passes over the fixed pulley, d^ and thence over the fixed triple block, B, and the moveable triple block, P, below which the load, L, hangs. Now, if the radius of the handle be 6 times that of the pinion, the radius of the wheel W 10 times that of iu axle, and a power equivalent to 30 lbs. be exerted at a; then, since a triple moveable pulley gives a mechanical advantage of 6 to 1, we shall have 30x6xl0x6» 10800 lbs. and such would be the load, L. that might be raised by a powe applied at a, were it not for the loss occasioned by friction. HI.] MBCHANICAL POWERS. 247 making such a calculation, the subjoined table, exhibiting at one the ratio of the power to the weight, in all the simple mechani- wers, will be of service. ipcioo oi Power* and axle . id wheels §:— tem (fig. 199) lo. (fig. aoo) to. (fig. 201) 9d plane Ratio of P : W. ■■^7 i.S. r n 1 :n 1 :^- 1 : 2--1 Symbol L equals the leverage of the power, / that of the weight. I Resrad. of the wheel, { r that of the axle. \ N •» No. of teeth in I the wheel, n the No. [ in the pinion, ) n equals the No. of > pulleys, both fixed ) and moveable. IX =» the length of the plane, h its height. Cx — the length of the J side of the wedge, b i the thickness of its Chack. ^ e<a the drcumferenoe V of the circle described J by the power, and d /the vertical disUnce ^between two threads. ObMrvation. / In all the cases both , the friction and weight j of the machines them. ^ selves are neglected, The strings are all supposed to be pa- raUel. When the power acts in a direction parallel to the plane. f When the resistance J inperpendiculartothe y side of the wedge, the (, wedge being single. FABT II CEAF. IT. 5«rT. L Gttfffrm DrfimiTm m M . Er3^!»T^r:'3 cniL7nHe» nie -Sacszae of sLe pumme and the ~ rivsL ic itin-^iassx f iba&. a» v^iicr, aerrair, &c^ mnd that of 2. A ffwfKt jtmmi s a Vocj vintfe parts are rerr miDOte, vieldiag to acT f!:«Te lAzrewvc XT«aK h . Xiov«icr saail), uid bj ao jielding srre Mifff^mr t ira l ideas of a flaid bodr, bj com j : h %o a beap of a^ ; b«t die xBiTkwiabuitj of ^^^^ flaiditv b? ii^p^i aaj \jzA of 3Kc£u3caJ cosauaiitaoii. vill apfwar br eonsidenii^ twci ■ wi 9i tibe cRsaMcaaen ttpcrwrr to coofdtate a fluid bodj : I . That^i^=t tbe parts. iMKvidataDdin* aar compmaoii. maj be mored in tioa to eadk oiLer, aiih tbe smallest coooeiTabie foree, or will gire DO mnuMt retu^mmct to mocioo wiibio the mass in anj directioD. 2. That the parts shall sravitate to each ocber, wfaerebr there is eoostant teodencT to amnse tbemselTes about a common center, i form a j^pKerical bodv ; which, as the parts do not resist motion, i^^^ easilj effected in soiali bodies. Henee the appearance of drops^^ alurajs takes p!ace when a jlmiJ is in proper circamstances. It i^^^ obrions that a body of sand can bj no means conform to thiMJiii^ drciraistances. Differeot fluids hare different degrees of flaiditr, according to th^^ ^KiIitT with which the particles maj be mored amongst each other. — Water and mercary are classed anumg the most perfect floids— ^ Many floids hare a rcrr sensible degree of tenacity, and are therefor^^' called viscous or imperfect fluids. 3. Fluids may be diTided into compressible and ineompressMe^ otf^ elastic and non-elastic fluids. A compressible or elastic fluid is on^ whose apparent magnitude is diminished as the pressure upon it vm increaseil, and increued by a diminution of pressure. Such is air.^ and the different Tapours. An incompressible or non-dastie flui<3 (called also a liquid) is one whose dimenuons are not sensibl/ affected by any augmentation of pressure. Water, mercury, oil, &c ^ are generally ranged under this class. It has been of late years proposed to limit the application of the term fluidi to those which are ekutiCy and to apply the word liquid to such as are non-elastic. CHAP. IV.] PKKSSURB OP FLUIDS. 249 4. The specific gravity or density of any solid or flaid body, is the absolute weight of a known volume of that substance ; namely, of that volume which we take for unity in measuring the capacities of bodies. Sbct. II. Pressure and Equilibrium of Non-elastic Fluids. 1. Fluids press equally in all directions^ upwards, downwards, aslant, or laterally. This constitutes one essential difference between fluids and solids, solids pressing only downwards, or in the direction of gravity. 2. The upper surface of a gravitating fluid at rest is horizontal. 3. The pressure of a fluid on every particle of the vessel containing it, or of any oUier surface, real or imaginary, in contact with it, is equal to the weight of a column of the fluid, whose base is equal to tliat particle, and whose height is equal to its depth below the upper BDifaoe of the fluid. 4. If, therefore, any portion of the upper part of a fluid be Replaced by a part of the vessel, the pressure against this from below niil be the same which before supported the weight of the fluid i^emoTed, and every part remaining in equilibrium, the pressure on the bottom will be the same as it would be if the vessel were a prism Or a cylinder. 5. Hence, the smallest given quantity of a fluid may be made to Produce a pressure capable of sustaining any proposed weight, either oy diminishing the diameter of the column and increasing its height, Or hy increasing the surface which supports the weight. 6- The perpendicular pressure of a fluid on any surface, whether >rertica], oblique, or horizontal, is equal to the weight of a column of ^e fluid whose base is equal to the surface pressed, and height equal ^o the distance of the center of gravity of that surface below the Opper horizontal surface of the fluid. 7. Fluids of different specific gravities that do not mix, will ^soanterbalance each other in a bent tube, when their heights above tihe surface of junction are inversely as their specific gravities. A portion of fluid will be quiescent in a bent tube, when the upper surface in both branches of the tube is in the same horizontal plane, or is equidistant from the earth's center. And water poured down one branch of such a tube (whether it be of uniform bore throughout, or not) will Ti»e to its own level in the other branch. Thus, water may be conveyed by pipes from a spring on the side of a hill, to a reservoir of equal height on another hill. 8. The ascent of a body in a fluid of greater specific grarity than itself, arises from the pressure of the fluid upwards against the under surface of the body 9. The center of pressure is that point of a surface against which any fluid prestes, to which if a force equal to the whole pressure 250 PRESSURE OF FLUIDS. | were applied it would keep the surface at rest, or balance ency to turn or move in any direction. 10. If a plane surface which is pressed by a fluid be pn the horizontal surface of it, and their common intersection be as the axis of suspension, the centers of percussion and of will be at the same distance /rom the axis, 11. The center of pressure of a parallelosram, whose upp in the plane of the horizontal level of the liquid, is at § oj (measuring downwards) that joins the middles of the two } sides of the parallelogram. 12. If the base of a triangular plane coincides with tl surface of the water, then the center of pressure is at the i the line drawn /rem the middle of the base to the vertt triangle. But, if the vertex of the triangle be in the uppe of the water, while its base is horizontal, Ae center of pt at ^ of the line drawn from the vertex to bisect the base. Id. If in any closed vessel containing a fluid suppose without weight, an opening or orifice be made and any pn applied, that pressure will be equally distributed over tl interior surface of the vessel; and if the fluid has a wei[ own, the pressure upon any point will equal the sum of tl buted pressure and the pressure occasioned by the weight of at that point. ILLUSTRATIONS AND APPLICATIONS. t. If several glass tubes of different shapes and sizes be put into a larger glass vessel containing water, the tubes being all open at top ; then the water will be seen to rise to the same height in each of them, as is marked by the upper surface a c, of the liquid in the larger vessel. 2. If three vessels of equal bases, one cylindrical, the sec siderably larger at top than at bottom, the third considerab top than at bottom, and with the sides of the two latt< regularly or irregularly sloped, have their bottoms moveable, close by the action of a weight upon a lever ; then it will I that when the same weight acts at the same distance upon t water must be poured in to the same height in each vessel I pressure will force open the bottom. 3. Let a glass tube open at both ends (whether cylindrio does not signify) have a piece of bladder tied loosely over < so as to be capable of hanging below that end, or of rising u it, when pressed from the outside. Pour into this tube son tinged red, so as to stand at the depth of seven or eight inci CHAP. IV.] bramah's PBESS. 251 then imroerae the tuhe with its coloured water vertically into a larger glass vessel nearly foil of colourless water, the hladder heing down-^ wards, serving as a fiexihle bottom to the tube. Then, it will be observed that when the depth of the water in the tube exceeds that in the larger vessel, the bladder will be forced behw the tube, by the excess of the interior over the exterior pressure : but when the ex- terior water is deeper than the interior, the bladder will be thrust up within the tube, by the excess of exterior pressure : and when the water in the tube and that in the larger vessel have their upper surfaces in the same horizontal plane, then the bladder will adjust itself into a fiat position, just at the bottom of the tube. The success of this experiment does not depend upon the actual depth of the water in the tube, but upon the relation between the depths of that and the exterior water; and proves that in all cases the deeper water has the greater pressure at its bottom, tending equally upward or downward. 4. The hydrostaticcU paradaof^ as it is usually denominated, results Prom the principle that any quantity of a non-elastic fluid, however ma]}, may be made to balance another quantity; or any weight, as arge as we please (§ 5, page 249). It may be illustrated by a naohine, the hydrostatic heUow9^ which consists »f two thick boards DC, F£, each about 16 r 18 inches diameter, more or less, covered r connected firmly with leather round the dges, to open and shut like a common bel- yy/v9y bot without valves; only a pipe A B, boat 8 feet high, is fixed into the bellows boTO F. Now, let water be poured into lo pipe at A, and it will run into the bellows, radoally separating the boards by raising the pper one. Then, if several weights, as three hundred-weights, be lid upon the npper board, and water be poured in at the pipe till it » fnJl, it will sustain all the weights, though the water in the pipe bonld sot weigh a quarter of a pound : for the pipe or tube may be a sniall as we please, provided it be but long enough, the whole flTect depending upon the height, and not at all on the width of the ipc, for the proportion is always this: — As the area of the orifice of the pipe is to the area of the bellows board, so is the weight of water in the pipe, above D C, to the weight it will sustain on the board. 5. In lieu of the bellows part of the apparatus, the leather of which rould be incapable of resisting any very considerable pressure, the ale Mr. Joseph Bramah used a very strong metal cylinder, in which I piston moved in a perfectly air and water tight manner, by passing .hroogh leather collars, and as a substitute for the high column of ivaler be adopted a very small forcing pump, to which auy power can be applied ; M>d thus the pressing column becomes indefinitely long. 252 biumah's press. [part ii. although the whole apparatus is very compact and takes hut little room. Figure 207 is a section of one of tliese presses, in which t is the piston of the large cylinder, formed of a solid piece of metal turned truly cylindrical, and carrying the lower hoard v of the press upon it : u is the piston of the small forcing pump, being also a cylinder of solid metal moved up and down hy the handle or lever w. The whole lower part of the press is sometimes made to stand in a case XX, containing more than sufficient water, as at y, to Bll both the cylinders; and the suction pipe of the forcing pump u dip])ing into this water will be constantly supplied. Whenever, therefore, the handle w is moved upwards, the water will rise through the conical metal valve jt, opening upwards into the bottom of the pump u ; and when the handle is depressed that water will be forced through another similar valve a, opening in an opposite direction in the pipe of communication between the pump and the great cylinder ft, which will now receive the water, by which the piston rod t will be elevated at each stroke of the pump «. Another small conical valve c is applied by means of a screw to an orifice in the lower part of the large cylinder, the use of which is to release the pressure whenever it may be necessary ; for, on opening this valve, any w^at ei^ which was previously contained in the large cylinder ft, will run off into the reservoir y by the passage d^ and the* piston t will descend ^^ so that the same water may be used over and over again. Th( power of such a machine is enormously great; for, supposing th hand to be applied at the end of the handle tr, with a force of onl 10 pounds, and that this handle or lever be so constructed as t multiply that force but 5 times, then the force nith which the pistoi u descends will bo equal to 50 pounds: let us next suppose that th< magnitude of the piston t is such, that the area of its horizontc section shall contain a similar area of the smaller piston u 50 times then 50 multiplied by 50 gives 2500 pounds, for the force wit which the piston i and the lower board v of the press will rise. J man can, however, exert ten times this force for a short time, am could therefore raise 25,000 pounds ; and would do more if a greai disproportion existed between the two pistons / and u^ and the lev w were made more favourable to the exertion of his strength. This machine not only acts as a press, but is capable of niai other useful applications, such as a jack for raising heavy loads, even buildings ; to the purpose of drawing up trees by their rooi or the piles used in bridge-building. To find the thickness of the metal in Bramah's press, to resi certain pressures, Mr. Barlow gives this theorem, t = — — whe J p =s pressure in lbs. per square inch, r = radius of the cylinder, i == its thickness, and c = 1 8000 Ihs. the cohesive power of a squ^^y inch of cast iron. £a;. Suppose it were required to determine the thickness of m^tsT in two presses, each of 6 inches radius, in one of which the pressure I CHAP. IV.] FRBSSURS AGAINST SLUICE-GATES, ETC. 253 may extend to 4278 pounds, in the other to 855G pounds per square inch. Here in the first case, 4278 X 6 .««,., t . , ' = ,»^^^ 7x=^ = 1*87 inches, thickness. 18000 — 4278 In the second, 8556 X 6 . .o . L 11 ^ = .^^^^ TTz-zz^ = 5*43 inches, thickness. 18000 — 8556 ' The usual rules, explained helow (Art. 10), would make the latter thickness douhle the former : extensive experiments are necessary to tell whicli method deserves the preference. 6. If b equals the breadth, and d the depth of a rectangular gate, or other surface exposed to the pressure of water from top to hottom ; then the entire pressure is equal to the weight of a prism of water inrhose content is ^ hd^. Or, if ^ and d be in feet, then the whole pressure = 31 J ft d\ in lbs., or nearly = -y\ b d\ in cwts. 7. If the gate be in the form of a trapezoid, widest at top, then, i f B and b be the breadths at the top and bottom respectively, and d the depth, the whole pressure in lbs. = 31 J {^^ (B — b) + b} d^ whole pressure in cwts. = -^ {^^ (B — b) -^ b}d^ nearly. 8. The weight of a cubic foot of rain or river water, is nearly equal ^o -,\ cwt. The pressure on a souare inch, at the depth of THiR-ty feet is very nearly THiR-/^it pounds. The pressure on a square foot is nearly a ton at the depth of thirty- six feet. [The true depth is 35-84 feet.] The weight of an ale gallon of rain water is nearly 10| lbs., that of an imperial ^Woii 10 lbs. The weight of a cubic foot of •ea-water is nearly ^ of a cwt. These are all useful approximations ; the actual weight of a cubic foot of distilled water is 6*25 lbs. Thus, the pressure of rain water upon a square inch at the depth of 3000 feet, is 1300 lbs. And tlie pressure upon a square foot at the depth of 108 feet is nearly three tons. 9. In the construction of dykes or embankments, both faces or slopes should be planes, and the exterior and interior slopes should make an angle of not less than OO"*. For, if A D' (fig. 208) be the exterior slope, and the angle D^ A B be acute, E D' perpendicular to A B is the direction of the pressure upon it; and the portion D'A E will pro- bably be torn off. But when DA is the exterior face, making with ABan obtuse angle, the direction of the pressure falls within the base, and therefore augments its stability. 10. The strength of a circular bason confining water requires the consideration of other principles. 254 FLOATING BODIES. [PART II. The perpendicular pressure against the wall depends merely on the altitude of the fluid, without being affected by the volume. But, as Professor Leslie remarks, the longitudinal effort of the thrust, or its tendency to open the joints of the masonry, is measured by the radios of the circle. To resist that action in very wide basons, the range or course of stones along the inside of the wall must be proportion- ally thicker. On the other hand, if any opposing surface present some convexity to the pressure of water, the resulting longitudinal strain will be exerted in closing the joints and consolidating the building. Such reversed incurvation is, therefore, often adopted in the construction of dams. Upon similar principles, the thickness of pipe$ to convey water, must vary in proportion to — , where h is the height of the head of water, d the diameter of the pipe, and c the measure of the cohe«on of a bar of the same material as the pipe, and an inch square. A pipe of cast iron^ 15 inches diameter, and \ of an inch thick, will be strong enough for a head of 600 feet. A pipe of (Hik of tlie same diameter, and 2 inches thick, woold sustain a head of 180 feet. Where the cohesion is the same, t varies B&hd; or as HD : T :: : hd : t^m the comparison of two cases *. Example, What, then, must be the respective thicknesses of pipes of cast iron and oak, each 10 inches diameter, to carry water from sh head of 360 feet ? Here, 1st, for cast iron: HD (= 600 X 15) : T (= ^) :: A rf (= 860 x 10) 860 X 10 X S 10800 s r • ,_ eOOT"! 5'ir4 = 36000 = ^^ ^^ *^ '°^*^- 2ndly, for oo^: HD (a= 180 X 16) : T ( = 2) :: A J (= 360 x 10) 860 X 10 X ft ^, „ «« . , -i8o-^ri5- = ^ = *=^^^°^^^ Sect. III. Floating Bodies. 1. If any body float on a fluid, it displaces a quantity of the fluid equal to itself in weight. * To ascertain whether or not a pipe is strong enough to sustain a proposed pressure, it is a good custom amongst practical men to empk>y a «^l|^e«lo#, usually of an indi in diameter, and load it with the proposed weight, and s surpluM determined by practice. Then, if the proposed pressure be aj^ied interiorly, by a forcing pump, or in any other way, if the pipe remain sound in all its parts after the safety-valve has yielded, such pipe is regarded as sufficiently strong. The aetual pressures upon a pipe of any proposed diameter and head, may evidently be determined by a similar method. CHAP. IV.J PARKY S FLOOD-OATE. 255 '2. Also, the centers of gravity of the body and of the fluid di8- pkieed, must, when the body is at rest, be in the same yertical line. 8. If aressel contun two fluids that will not mix (as water and mercurj), and a solid of some inteimediate specific gravity be im- mened under the surface of the lighter fluid and float on the heavier; the part of the solid immersed in the heavier fluid, is to the whole solid aB the difference between the spedflc gravities of the solid and the lighter fluid, is to the difference between the specific gravities of the two fluids. 4. The buoyancy of casks, or the load which they will carry with- out sinking, may be estimated by reckoning 10 lbs. avoirdupois to the ale gallon, or 8| lbs. to the wine gallon. 5. Tlie buoyancy of pontoons may be estimated at about kalf a kwired-wei^ for each cubic foot. Thns a pontoon which contained 06 cubic feet, would sustain a load of 48 cwt. before it would sink. This is an approximation, in which the difference between -^ and |, that is, ^^ of the whole weight, is allowed for that of the pontoon itself. 6. The principles of buoyancy are very ingeniously applied in Mr. Farcy's tdf -acting flood-gate. In the case of common sluices to a niil-dam, when a sudden flood occurs, unless the miller gets up in the nip;ht to open the gate or gates, the neighbouring lands may be* come inundated ; and, on the contrary, unless he be present to shut th«n np when the flood subsides, the mill-dam may be emptied and the water lost which he would need the next day. To prevent either of these occurrences, Mr. John Farey, whose talent and ingenuity are well known, has proposed a ielf-acting flood-gate^ the following de- scription of which has been given in the Mechanics Weekly Journal. — j _. . =^-^IS w^ ^^ r^^ _B gj:^::x Fi ,..,,..,^„,.„„.,,...jp.-., — — wL-t^ m~r V - - m <i ter '^ t^ .^ ,^i^^V^ — ^ — ^^^— -- — ^ " — 1 SA / " IV :/ 1 ,_>^J:^-^- 1 4 \y ! \ V^-i^p^ — r p [:■ '*~^ 256 SPECIFIC GRAVITIES. [PART II. A A represents a vertical section of a gate poised upon a horizon- tal axis passing rather above the center of pressure of the gate, so as to give it a tendency to shut close: a a is a lever, fi.xed perpendicu- lar to the gate, and connected by an iron rod with a cask, 5, which is floated whenever the surface of the water rises to the line B, D, which is assumed as the level of the wear, or mill-dam, B,C,£, F, in which the flood-gate is placed : by this arrangement it will be seen that when the water riscK above the dam, it floats the cask, opens the gate, and allows the water to escape until its surface subsides to the proper level at B, D; the cask now acts by its weight, when unsupported by the water, to close the gate and prevent leakage. The gate should be fitted into a frame of timber, H,K, which is set in the masonry of the dam. The upper beam H, of the frame being just level with the crown of the dam, so that the water runs over the top of the gate at the same time that it passes through it: to prevent the current disturbing the cask, it is connected by a small rod, e, at each end, to the upper beam, H, of the frame, and jointed in such a manner as to admit of motion in a vertical direction. 7> By means of the same principle of buoyancy it is, that a hollow ball of copper attached to a metallic lever of about a foot long, is made to rise with the liquid in a water-tub, and thus to close the cock and stop the supply from the pipe, just before the time when the water would otherwise run over the top of the vessel. 8. This property, again, has been successfully employed in pulling up old ])i]e8 in a river where the tide ebbs and flows. A barge of considerable dimensions is brought over a pile as the water begins to rise: a strong chain which has been previously fixed to the pile by a ring, &c. is made to gird the barge and is tlien fastened. As the tide rises the vessel rises too, and by means of its buoyant force draws up the pile with it. In an actual case, abarge 50 feet long, IS feet wide, 6 deep, and draw- ing *2 feet of water, was employed. Here, 50 x 12 x (6 — a) x ? = ^Q X ^^ X 16 ^ ^^^ X 7| = 1344 + 274 = 1371f cwt.= 66J tons nearly, the measure of the force with which the barge acted upon the pile. Sect. IV. Specific Gravities. 1. If a body float on a fluid, the part immersed is to the whole body, as the specific gravity of the body to the specific gravity of the fluid. Hence, if the body be a square or a triangular prism, and it be laid upon the fluid, the ratio of that portion of one end which is im- mersed, to the whole surface of that end, will serve to determine the specific gravity of the body. 2. If the same body float upon two fluids in succession, the parts immersed will be inversely as the specific gravities of those fluids. CRAP. ly.] SPECIFIC GRAVITIES. 257 3. The weight which a body loses when wholly immersed in a fluid is equal to the weight of an equal bulk of the fluid. When we say that a body loses part of its weight in a fluid, we do not mean that its abtoluie weight is less than it was before, but that it is partly supported by the reaction of the fluid under it, so that it requires a less power to sustain or to balance it. 4. A body immersed in a fluid ascends or descends with a force equal to the difference between its own weight and the weight of an eqoa] bnlk of fluid ; the resistance or viscidity of the fluid not being considered. 5. To Jind ike specific aravity of a fluids or of a solid. — On one arm of a balance suspend a globe of lead by a fine thread, and to the other fasten an equal weight, which may just balance it in the open air. Immerse the globe into the fluid, and observe what weight balances it then, and consequently what weight is lost, which is proportional to the specific gravity as above. And thus the proportion of the Q)ecific gravity of one fluid to another is determined by immersing the globe successively in all the fluids, and observing the weights lost in each, which will be the proportions of the specific gravities of the ^mds sought. This same operation determines also the specific gravity of the Solid immerged, whether it be a globe or of any other shape or bulk, Supposing that of the fluid known. For the specific gravity of the lluia is to that of the solid, as the weight lost is to the whole ^-eight. Hence also may be found the specific gravity of a body that is lighter than the fluid, as follows : O. To find the specific gravity of a solid that is lighter than the fluid y ^i^stPOieTj in which it MjEm/.— Annex to the lighter body another that i^ much heavier than the fluid, so that the compound mass may sink in tbe fluid. Weigh the heavier body and the compound mass sepa- «-«tel J, both in water and out of it ; then find how much each loses ivi urater, by subtracting its weight in water from its weight in air; ^nd subtract the less of these remainders from the greater. Then, As this last remainder, : the weight of the light body in air, : : the specific gravity of the fluid, : the specific gravity of that body. 7. The specific gravities of bodies of equal weight are reciprocally proportionar to the quantities of weight lost in tbe same fluid. And tience is found the ratio of the specific gravities of solids, by weighing in the same fluids masses of them that weigh equally in air, and noting the weights lost by each. 8. Instead of a hydrostatic balance^ a hydrostatic steelyard is now frequently employed. It is contrived to balance exactly by making the shorter end wider, and with an enlargement at the extremity. The shorter arm is undivided, but the longer arm is divided into short equal divisions: thus, if that longer arm be 8 inches long, it 258 CAPILLARY ATTRACTION. [PART II. may be divided into 400 parts, the divisions wmmeneing at A (fig. 209V Then, in using this instrument, any convenient weight is suspended by a hook from a notch at the end of the scale A. The body who$e specific gravity is to be determined, is suspended from the other am^^ by a horse-hair, and moved to and fro till an equilibrium is produced^ _ , Then, without altering its situation at D on the beam, it is immenev:^ in water, and balanced a second time by sliding the couDterpois^^^ from A, say to C. Here, evidently, the weight in water : weight in air : : B C : B Am^ ^ and the loss of weight in water : weight in air :: AG : AB. ^ , weight in air AB .^ Consequently, — — = — -- = specific gravity. Joss A C With such an instrument nicely balanced upon a convenient ped^^^ tal, I find that the specific gravities of solids are ascertainable b^^n^^ with greater facility and correctness than with any hydrostaH^t^^ balance which I have seen *. A copious table of the specific gravities of various substances "99111 be found in the Appendix, Table X. Sect. V. On Capillary Attraction, 1 . If two plates of glass be set up vertically in a shallow vessel, containing a coloured liquid, in such a way as to touch along the edges AB, fig, 210^ and forming a very acute anele vidth each other, the coloured liquid will rise between the two plates to a height in- versely as the distance between them ; from which it follows, that its surface will form a rectangular hyperbola, whose two asymptotes are the edge of the plate A B and the surface of the li(^uid. And if a glass tube, the bore of which is small, be immersed in the liquid, it will rise within the tube to a greater height than the surfiEMse of the liquid without the same. 2. The cause of this phenomenon is an attractive force existing between the liquid and the glass, and which is termcfd capillary at- traction, from capiUuSy the Latin for hair, because it only Utkes place in tubes, the bores of which are small, resembling hairs. 8. The heights to which fluids rise in capillary tubes are inversely as their diameters : and they are found to nse to the same height in a vacuum, as in the air. 4. In a tube -r^n^^ ^^ ^^ ^^^^ ^° diameter, water rises 5*3 inches; and as the height is inversely as the diameter, their product will be a constant quantity, therefore 5*3 X *01 = *053 = ^, which may be taken to represent the attractive force between water and glaaa. The following are the values of ^, according to different experimenters :— - * We owe this coDtrivance to Dr. CotUtiy of Philaddpfaia. CHAP. IV.] CAPILLARY ATTBACTION. 259 Sir I. Newton -020 Sir D. Brevv-ster -033 M.M.Gay Lussac -046 Mr. Atwood '053 5. The height at which the fluid is supported hy capillary attrac- tion in a tube of Yar3dng diameter, is that due to its size at the sur- face of the liquid, without any regard to the dimensions or shape of the lower portion. Thus, in a vessel of the form shown in figure 211, terminating in a capillary tube, the liquid will be supported at the same heifi;ht in this tube (and also fill the whole of the space abc of the TesscI) as if the tube had been of uniform dimensions throughout its whole length, and every where equal to its upper portion. The water, however, contained in the lower portion of the vessel a & c, is in this case supported by the pressure of the atmosphere, as it ceases to be supported in a vacuum. 6. Different liquids rise to different heights in capillary tubes of the aaxne bore, depending upon the attraction between them and ^lass. The following are Dr. Brewster's results for several fluids, irith a tube '0561 inches in diameter : — Height of Value of the aaoent in inchef. constant q. Water -587 0327 Water, very hot 537 0301 Muriatic acid 442 0248 Nitric acid -395 -0222 Spermaceti oil -392 0220 Olive oil -387 '0215 Oil of turpentine -333 -0187 Alcohol -317 -0178 iEther -285 -0160 Sulphuric acid 200 0112 7. The internal diameter of a uniform capillary, or other small tube, may be found in the following manner. Let the tube be weighed when empty, and again when filled with mercury, and let w be the difference of those weights in troy grains, / the length of the tube in inches; and d its diameter, then d = '019252 \/-j Thos, if the difference of the weights were 500 grains, and the length of the tube were 20 inches: we should have d = *010252 y^^ = -019252 X 5 = -09626 of an inch* * Tlie same thing may easily i>e aeoomplithed thot t — Let a cone of box wood, 0r of bran, be very aoeunitely turned, or about 6 inches in length, and the dia- naecer of its base aboat a quarter of an inch ; and let its curve surface i>e very aecoratelj marked with a series of parallel rings, about a twentieth of an inch aa i mde r, from its vertex to its base. Insert this cone carefully in the cylinder (ao that their axes shall coincide) as in fig. 212 : then it will be as V A i Va : t A B : eft; where, as the ratio of V A to V a is known by means of the equi- distant rings on the iorfaea, and A B is known, a b becomes determined. 82 260 SPFLUBNC£ OF FLUIDS. [pABT II. CHAP. V. HYDRODYNAMICS. HydrodynamicB is that part of mechanical science which relates to the motion of non-elastic fluids, and the forces with which they act upon bodies. This branch of mechanics is the most diflicult, and the least ad- vanced : whatever we know of it is almost entirely due to the re- searches of the modems. Could we know with certainty the mass, the figure, and the num- ber of particles of a fluid in motion, the laws of its motion might be determined by the resolution of this problem, viz. to find the motion of a proposed system of small free bodies acting one upon the other in obedience to some given exterior force. We are, however, ▼cry far from being in possession of the data requisite for the solution of this problem. We shall, therefore, simply present a few of the most usually received theoretical deductions ; and then proceed to state those rules which have resulted from a judicious application of theory to experiment. Skct. I. Motion and Effluence of Liquids. 1 . A jet of water, issuing from an orifice of a proper fonn, and directed upwards, rises, under favourable circumstances, nearly to the height of the head of water in the reservoir; and since the particles of such a stream are but little influenced by the neighbouring ones, they may be considered as independent bodies, moving initially with the velocity which would be acquired in falling from the height of the reservoir. And the velocity of the jet will be the same whatever may be its direction. 2. Hence, if a jet issue horizontally from any part of the side of a vessel standing on a horizontal plane, and a circle be described having the whole height of the fluid for its diameter, the fluid wiU reach the plane at a distance from the vessel, equal to that chord of the circle m which the jet initially moves. Thus, if AS (fig, 213) be the upper surface of the fluid in the ves- sel, B the place of the orifice, CF the horizontal plane on which the fluid spouts, then CF is equal to £D, the horizontal chord of the BPFLUXNCE OF FLUIDS. 261 le diameter is A C, passing throagh B. It is therefore evi- tbe orifice from which the fluid will spout to the greatest I situate at G, half the height of the fluid, and also, that if ires be made at equal distances ahoTC and below 6, the jet om both will strike the plane C £ in the same point. m a cylindrical or prismatic vessel empties itself by a small 3 Telocity at the surface is uniformly retarded; and in the nptying itself, twice the quantity would be discharged if it I full by a new supply. the quantity dischsrged is by no means equal to what would lole orifice, with this velocity. If the aperture is made sim- lin plate, the lateral motion of the particles towards it tends i the direct motion, and to contract the stream which has ifice, nearly in the ratio of two to three. So that, in order le quantity discharged, the section of the orifice must be to be diminished from 1 00 to 62 for a simple aperture, to ipe of which the length is twice the diameter, and in other »rding to circumstances. m. a syphon, or bent tube, is filled with a fluid, and its mersed in the fluids of different vessels, if both surfaces of are in the same level, the whole remains at rest ; but if the longer column of fluid in the syphon preponderates, ■easure of the atmosphere forces up the fluid from the higher til the equilibrium is restored ; and the motion is the more be difference of the levels is greater: provided that the ^ht of the tube above the upper surface be not more than loise to the pressure of the atmosphere. le lower vessel be allowed to empty itself, the syphon will mining as long as it is supplied from the upper, and the it descends the further below the vessel. In the same e discharge of a pipe, descending from the side or bottom i vessel, would be increased almost without limit by length- notch or sluice in form of a rectangle be cut in the ver- of a vessel full of water, or any other fluid, the quantity loagb it will be ^ of the quantity which would flow through nifice, placed horizontally at the whole depth, in the same reasel being kept constantly full. ivr«Deiit in the oonstniction of the syphon has been lately proposed pnv Mechanics^ Magazine^ and by M. burUem at Paris. It might mtageouftly used if constructed on a larse scale, for lowering the B-4laiBS or canals. The improvement in toe present syphon is, that pipe is enlarged to the same diameter as that of the syphon, and lened out to something of a funnel shape, as in fig. 214. In J action, the short arm is immersed in the water as in the usual bottom of the long arm is closed, the exhausting pipe is then filled hjr the funnel-^aped mouth. On the bottom of the long arm being Mtter flows out, and exhausts the air from the syphon, when the rh wished to be emptied flows out in a continual stream. z& i 262 PIPB8 AND CANALS. [PART II. 8. If a short pipe elevated in any direction from an aperture in t condnit, throw the water in a parabolic curre to the distance or range R, on a board, or other horizontal plane passing through the orifice^ and the greatest height of the spouting fluid above that plane, be H, then the height of the head of water above that conduit pipe, may be found nearfy: vis. by taking 1st, 2 cot £ = ^--; and 2ndly, the alti- 2 n tude of the head A = i R x cosec 2 E. Example. Suppose that R = 40 feet, and H = 18 feet. Then — == ~ = Mllllll =2cot60*'57': and A = i R x cosec2E = 20 X cosec 121* 5V = 20 x 1-177896 = 23-55792 feet, height required. Note. This result of theory will usually be found about ^ of that which is furnished by experiment Sbci. II. Motion of Water in Conduit Pipes and Open Canals^ over Weira^ 4^. 1. When the water from a reservoir is conveyed in long horizontal pipes of the same aperture, the discharges made in equal times are nearly in the inverse ratio of the square roots of the lengths. It is supposed that the lengths of the pipes to which this rule is applied, are not very unequal. It is an approximation not deduced from principle, but derived immediately from experiment. [Bossut, tom. ] 1, $ 647, 648. At § 673, he has given a table of the actual discharges of water-pipes, as far as the length of 2340 toises, or 14,950 feet English.] 2. Water running in open canals, or in rivers, is accelerated iu consequence of its depth and of the declivity on which it runs, till the resistance, increasing with the velocity, becomes equal to the acceleration, when the motion of the stream becomes uniform. It is evident that the amount of the resisting forces can hardly be determined by principles already known, and therefore nothing remains but to ascertain, by experiment, the velocity corresponding to different declivities, and different depths of water, and to try, by multiplying and extending these experiments, to find out the law which is common to them all. The Chevalier Du Buat has been successful in this research, and has given a formula for computing the velocity of running water, whether in close pipes, open canals, or rivers, which, though it may be called empirum^ is extremely useful in practice. Principea d'Hydrauligue. Professor Robison has given an abridged account of this book, m his excellent article on Rivers and Water-worics, in the JEnejfdopcadia Britannica. Let V be the velocity of the stream, measured by the feet it moves over in a second ; R a constant quantity, vis., the quotient obtained CHAP, v.] PIPES AND CANALS, 263 by dividiog the area of the transverse section of the stream, expressed in square feet, by the boundary or perimeter of that section, minus the saperficial breadth of the stream expressed in linear feet. The mean velocity is that with which, if all the particles were to move, the discharge would be the same with the actual discharge. The line R is <^Ied by Dn Buat, the raditi$; and by Dr. Robison, the k^raulie mecm depth. As its affinity to the radius of a circle eeenifl greater than to the depth of a river, we shall call it, witli the former, the radiui qfihe section, LasUy, let 8 be the denominator of a fraction which expresses the slope, the numerator being unity, that is, let it be the quotient ob- tained by dividing the length of the stream, supposing it extended in ^ straight line, by the difference of level of its two extremities : or, -virhich is nearly the same, let it be the co-tangent of the inclination or slope. 3. The above denominations being understood, and the section, as -well as the velocity, being supposed uniform, we have When R and 8 are very great, ^ = »'{sn^.-4' ""''^^ ^"-^ The logarithms understood here are the hyperbolic, and are found by multiplying the common logarithms by 2*3025851 ; or more easily by the method described at page 94. The slope remaining the same, the velocities are as n/r — y^. The velocities of two great rivers that have the same declivity, are as the square roots of the radii of their sections. If R is so small, that n/R — -^j^ = 0, or R = -^j, the velocity will be nothing; which is agreeable to experience; for in a cylindric tube R =s ^ Uie radius ; the radius, therefore, equals two-tenths ; so that the tube is nearly capillary, and the fluid will not flow through it. The vdodty may also become nothing by the declivity becoming so small, that if - is less than — — —-, or than Vn^h of an inch to an English 8 600000' ;^ ^ mile, the water will have sensible motion. 4. Ib a river, the greatest velocity is at the surface, and in the middle of the stream, from which it diminishes towards the bottom 2S4 PIPES AND CANALS. [PABT II. and the sides, where it is least. It has been found by experiments that if from the square root of the Telocity in the middle of the stream, expressed in inches per second, unity be subtracted, the square of the remainder is the velocity at the bottom. Hence, if the former velocity be = 9, the velocity at the bottom c=r— 2v^t?4-l (HI.) 5. The mean velocity, or that with which, were the whole stream to move, the discharge would be the same with the real discharge, is equal to half the sum of the greatest and least velocities, as computed in the last proposition. The mean velocity is, therefore, =»— ^/c-f^ (IV.) This is also proved by the experiments of Dn Buat. 6. Suppose that a river having a rectangular bed, is increased by the junction of another river equal to itself, the declivity remaining the same ; required the increase of the depth and velocity. Let the breadth of the river = 5, the depth before the junction </, and after it d?; and in like manner, v and v^ the mean velocities before and after; then ^ , is the radius before, and b + 2d ' 6 + 2x the radius after, so o = r — , supposing the breadth of the river to be such, that we may reject the small quantity subtracted from R, SOT R * in formulce (I. and II.); and, in like manner, v^ = r— !-. Then, substituting for R and R^, we have ^^30T , f~hd S* ^ ^ cH- 2<^' 307 / vof and c. =--T- X A/ -— . Multiplying these into the areas of the sections 5 </ and ior, we have the discharges, viz., 307 hds/hd ^^ 307 hxs/hx Now the last of these is double of the former ; therefore, hx^hx 2hds/hd x" 4rf» >/^ -f 2rf' 'ft-h2« 64-2rf' and t TT-k ^ = 1 r-i J a cubic equation which can always be resolved by Cardan's rule, or by the approximating method given at page 82. As an example, let ft = 10 feet, and </ = 1, then x^ — J « = ^*>, and X = 1*4882, which is the depth of the increased river. Hence we have 1-488 x t>j = 2 r, and 1'488 : 2 : : «? : Cj, or t? : ©j : : 37 to 50 nearly. CHAP, v.] CANALS, RIVKBS, &C. 265 When the water in a river receives a permanent increase, the depth and the velocity, as in the example above, are the first things that are augmented. The increase of the velocity increases the action on the sides and bottom, in consequence of which the width is aaemented, and sometimes also, but more rarely, the depth. The velocity is thus diminished, till the tenacity of the soil or the hard- ness of the rock affords a sufficient resistance to the force of the ^water. The bed of the river then changes only by insensible degrees, and, in the ordinary language of hydraulics, is said to be permanent, though in strictness this epithet is not applicable to the course of any river. 7' When the sections of a river vary, the quantity of water s-emaining the same, the mean velocities are inversely as the areas of ^he sections. This must happen, in order to preserve the same quantity of discharge. (Pla^air's Outlines,) 8. The following table, abridged from Du Buat, serves at once to compare the surface, bottom, and mean velocities in rivers, according t:o the formul® (III. and IV.) VdocUies of Rivers. VRI/K31TY IN INCHES. VELOCITY IN INCHES. Sm&ce. Bottom. Mean. Snr&ce. Bottom. Mean. 4 1- 2-5 56 42-016 49-008 8 3-342 5-67 60 45-509 52-754 la 6071 9-036 64 49- 56-5 16 9- 12-5 68 52-505 60-252 20 12065 16-027 72 56-025 64-012 24 15194 19-597 76 59-568 67-784 28 18-421 23-210 80 63-107 71-553 32 21-678 26-839 84 66-651 75-325 36 25- 30-5 88 70-224 79112 40 28-345 34-172 92 73-788 82-894 44 31-742 37-871 96 77-370 86-685 48 35151 41-570 100 81- 90-5 52 38-564 45-282 9. The Icnowledge of the velocity at the bottom is of the greatest use for enabling us to judge of the action of the stream on its bed. Every kind of soil has a certain velocity consistent with the sta- bility of the channel. A greater velocity would enable the waters to tear it up, and a smaller velocity would permit the deposition of more movable materials from above. It is not enough, then, for the sta- bility of a river, that the accelerating forces are so adjusted to the 260 CANALS, RIVBBSy &C. [PART II. size and figure of its channel that the current may be in train : it most also be in equilibrio with the tenacity of the channel. We learn from the obseryations of Du Buat, and others, that a velocity of three inches per second at the bottom will jast begin to work upon the fine clay fit for pottery, and howerer firm and com- pact it may be, it will tear it up. Yet no beds are more stable than clay when the velocities do not exceed this: for the water soon takes away the impalpable particles of the superficial clay, leaving the particles of sand sticking by their lower half in the rest of the clay, which they now protect, making a very perma- nent bottom, if the stream does not bring down gravel or coarse sand, which will rub off this very thin crust, and allow another layer to be worn off; a velocity of six inches will lift fine sand; eight inches will lift sand as coarse as linseed ; twelve inches will sweep along fine gravel ; twenty-four inches will roll along rounded pebbles an inch diameter; and it requires three feet per second at the bottom to sweep along shivery angular stones of the size of an egg. (i2o6i- son on Rivers,) 1 0. In the elbow or bend of a river, the velocity is always greater near the concave than the convex side. 1 1 . The swell occasioned by the piers of a bndge, or the sides of a cleaning sluice which contract the passage by a given quantity, for a short length only of the channel, may be determined when the ver- tical section of the river and the velocity of the stream are known, in the following manner. Let v be the velocity of the stream, inde- pendently of the effect of the bridge, r the section of the river, and a the amount of the sections between the piers; let 2^, instead of being taken at 64j, be reckoned 58'6, to accord with the results of experimental contractions through arches of bridges, &c., and let $ be the slope of the bed of the river, or the sine of its angle with the horizon ; then Du Btiat (tom. i. p. 225) gives for the swell or rise (R) of the stream in feet, which will be occasioned by the ob* struction. -(3^0-) (GI-); •• ■<^' 12. The value of s will, of course, be different in different cases; but if we assume -^ or '05, as a mean value, it will enable us to compute and tabulate results, which, though they cannot be pre- sented as perfectly correct, may be regarded as exhibiting a medium between those that will usually occur; and will serve to anticipate the consequences of floods of certain velocities, when constrained to pass through bridges which more or less contract the stream. CHAP, v.] SWELLS 0C0A8I0NED BY BRIDGES. 267 TaUe of the Rise of Water occaeumed hy Piere of Bridges^ or other Contractions^. Amoum of otHtTiictiaafi f^inupved with th* vortical HetJon of the Hiver. MOth. a^iothi. 3-lOlhA 40)>thi. a-ioth». B-ltrthj. 7'imhi. a-ieth.. o^iotiii. 4 Pruporliotml Riie of W»tcr, in f«rt and decimals. feel. tbe^ rwt fwt. f«t. fc«t. fMt. feeu fMt. <K11S7 (M)377 ^HMlMr 0*[IS2 OS<H!t 0-3631 (1^78 l^gw mam *-os77 <]^)IW5 flisai OS 102 «':t'^#4« ivn^^ 1 1199S ia37a \i1\m \OTtUll«T7 J flex>di. ^NHT? IM|*4 0^1 in D-aiSlfl «^JfP7 IHlfiHT MSiM> 4i»MI 20-1 A>4 (H*70l i>18» frasTf mi^ {*^iti |-7iM«i 3't7&5 r-Tjan 3tm^> ft-UfiS (^'^709 i^sim fl'STUS i*4aafl 2-6<ifpfi 5i>if>2 il'ftlff- 4ft- 1555 > Violent 1 flfxirti. criAsa 0^19 v\m iijyi?5 :!-Mifiti d7i^ iMtlH tts-7sie tP-i«7« C^4<m: (H»iSl i-57a> sri579 4-6511 a-SL^iTe n-mm Ji7'TiW> ri-iii78 O-eiia 11BB4 iteso 3*25* aawT H^AIM ^Mm li3tM±i I t'lJiit'iiPilly 033iO ^"'N>54 IMi^JO 2-55Ge A'W^ 7'MT3 14^4777 34a!J4e Ul-JSAi J. vjolcnt 10 0^41 19 \mn i-mj^ ^'li^ie s^asw 9^tJt» J77M1 4a*i+MJ iTk^um 1 floodi. , 13. It will be eyident, from an inspection of this table, that even in the case of ordinary floods, old bridges with piers and starlings, occupying 6 or 7-tentbs of the section of the river, will produce a swell of 2, 3, or more feet, often overflowing the river^s banks and occasioning moch mischief. Also, that in violent floods, an obstmc- tion amounting to T-tenths of the channel, will cause a rise of 7 or 8 feet, probably choking up the arches and occasioning the destruction of the bridge. Greater velocities and greater contractions produce a rapid augmentation of danger and mbchief ; as the table obviously shows. 14. The same principles and tabulated results «erve to estimate the fall from the higher to the lower side of a bridge, on account of an ebbing tide, &c. Thus, for old London Bridge, where the breadth of the Thames is 926 feet, and the sum of the water ways at low water only 236 feet, the amount of the obstructions was 690 feet, about 7| tenths of the entire section : so that a velocity of 3 1 feet per second would give a fall of nearly 4| feet, agreeing with the actual result. At Rochester Bridge, before the opening of the middle arches, the piers and starlings presented an obstruction of 7-tenths, and at the time of greatest fall> the velocity 100 yards above bridge exceeded 6 feet per second. This, from the table, would occasion a fall of more than 6-7 feet ; and the recorded results vary from 6| to 7 feet At Westminster Bridge, where the obstructiona are about l-sixth of the whole channel, when the velocity is 2^ feet, the fall but little exceeds half an inch: a result which the table would lead us to expect. •. A timilar table was computed by Mr. Wright of Durham, more than ftfty jmn ago, and inierted in the first edition of Dr. Hutton*B treatiM on Bndget ; but it it not constructed upon a correct theory. 268 eytblwbin's bbsults. [part u. 15. Mr. Eytelwein^ a German mathematiciany has devoted much time to inquiries in hydrod3mamic8. In his investigations he has paid attention to the mutual cohesion of the liquid moleculee, their adherence to the sides of the vessel in which the water moves, and to the contraction experienced by the liquid vein when it issues from the vessel under certain circumstances. He obtains formulse of the utmost generality, and then applies them to the motion of water; 1st, in a cylindric tube; 2ndly, in an open canal. 16. Let d be the diameter of the cylindric tube EF (fig. 215), h the total heieht F G of the head of water in the reservoir above the orifice F, and ? the length £ F of the tube, all in inches : and v equal the velocity in inches per second with which the fluid will issue from the orifice F ; then V • ^'iV'-f^ <"•> which, multiplied into the area of the orifice, will give the quantity discharged per second. 1 7. Let d = the diameter of the pipe in inches, Q = the quantity of water in cubic feet discharged through the pipe per minute, / = the length of the pipe in feet, and h = the difference of level between the surface of the water in the reservoir and at the end of the pipe, or the head; then, any three of these quantities being given, the fourth may be determined from the follo\idng formulae: — ^= ^'O-^^^Q-C^^^^-^^ (VII.) « == V^tt^^^^^t:^ (VIII.) hd' •0448 (/ + 4-2 d) I = ^^' , - 4 2rf (IX.) •0448 Q'* ^ ^ . -0448 Q2(/ + 4-2 rf) .„^ ^= ^^5 (^-^ These formulae are more convenient when expressed logarithmic- ally, and then become log (/ = J {2logQ +2-6515 + log(/ + 4-2(/) -logA} (XL) logQ = I {logh + 5\ogd-^ 2-6515 - log(/ + 4-2flO} ... (XIL) log / =log;i+5logrf-2'-6515-2logQ,[neglecting — 4-2flri(XIII.) log A = 2 logQ+ ■2-6515 + log (/ + 4-2 (/) - 5 log rf (XIV.) 18. When a pipe is bent in one or more places, then if the squares of the sines of the several changes of direction be added into one sum «, the velocity v will, according to Lang^dorf^ be foond by the theorem CHAP. v.] bttblwbin's bbsults. 269 = a/^ SMdk (XV.) ■^l.+ ids /, k^ d^ and 9, being all in inches. 19. For open canals. — Let v be the mean velocity of the current in feet, a area of the vertical section of the stream, p perimeter of the section, or snm of the bottom and two sides, / length of the bed of the canal corresponding to the fall hy all in feet : then * = /^ 9582 — + 00111 - 0-109 (XVI.) The experiments of M. Bidone^ of Turin, on the motion of water in canals, agree within the 80tli part of the results of computations from the preceding formulae. 20. For apertures in the sides or bottom of vessels,— \^ q equals the €]iiantity of water discharged in cubic feet per minute, v the velocity of the effluent water in feet per second, through the aperture, a the sreA of the aperture in square inches, and h the height from its center to the surface of the water ; we have v^Cs/h (XVII.) y=-4ie7ac ^h (XVIII.) In which c is a constant quantity depending upon the nature of the aperture, and the value of which for several different forms is con- tained in the following table. Nature of the Orifices employed. Ratio between the theoretical and real Coefficients for finding the velocities in Eng. feet. For the whole velocity due to the height For an orifice of the form of the contracted > vein. For wide openingi whose bottom is on a 1 kvd with that of the reservoir ; for sluices ( with walls in a line with the orifice ; for T bridges with pointed piers. For narrow openings whose bottom is on a^ levd with that of the reservoir; for mailer openings in a sluice with side - waDa ; for abrupt projections and square piers of bridges. J For openinp in sluices without side walls For sui orinoe in a thin nlate 1 to 100 1 to 0-969 1 to 0-961 1 to 0-861 1 to 0-635 1 to 0-621 1 to 0-510 1 to 0-671 1 to 0-808 80458 7 8 7-7 69 51 5-0 41 5-4 6-5 When there is a short cylindrical pipe pro- ) jeeting on the inside of the vessel, length ( 9 to 4 times the bore, and rim like con- f tracted vein, not fulL J Idem, when it runs fuU bore of water Idem, when it does not project inside the \ VCMCL S MOTION IN PIPI89 BTC. [PA&l r t -I Bore of the pipe in inches. '*"*'p^ :I?:?2'« oSTSToo « ^ ?.o»i^»«©-2 .SO 1- S 2 2 ZS ^ * *^* *® "* '* •* ** ** ** ** '^ '^ ^ '^ '^ 6 « © w lb r»b « ©• Oft » 00 r»^ « '^ ^ 1- © © ep -* «o © 00 ^© t^oo "* 00 cc* « »^r*9 t^^oot^««'<«eoeoc«G«e4p^i-«^-4^©©©c a. 1 a 1 1" © rt ©© *-©©-*(» rt ^ -5 (N :• »^ »-« -r- <?•« « 00 — ©to©6>o©ooeo^oo©<«^^©o»aor<»cs eo©©kfd^MMeoG«M*iMpi4^PH^©©©©©© .So 1- ©©ie'«eo©oi©e4e4eo-4<«-4^©or<»©ie© •^do©©^©« &i©69ib99<^©©0Dt^<b^«ib ©©«^oooi&«e«ei-«'^'^-H^©©©©©©© •0 S •3 > a CO fa .go fa t>.N©eO©iO'*0 — C0M^»O» — — '*00«9»« •^©o© ^©©r^ib&«©©©Qor«©>e<«'^ei»f t^-^eowe^N-^ — »- — — ©o©©oo©©© »^»o©P-©« o"** 10© M ©!'*«« — ©«»«r ©N-iy — r^'*e«©f^©oor*o «©'*'* eo«*ir ^0094^-4 — -«^©©©o©©©©©©© d«o 1- © CO 10 © t^O »^0D © © •?• © «p © •© W © t *:- ^ lb 91 ^ © 00 r«»^ •« kid ^ CO eo 99 e« o« -N .. OI^pHp-©©©©©©©©©©©©00 .So 1- »^— '*t>.©«eit-cp»'?''r'»^T*'?'9f ;ii,:^QO©ib'ii"^ooeoN©i«'^'-^-*'^-^< ^o©©©oo©©©©oo©ooe .S«o 10 © <N 00 10 CO - © © r^o « '^ "^ « o' ;,«©,^^;^^-©©©©©©©©< ©©©©©©oo©©o©©©e© Bore of the pipe 'in inches. '*"*'^ :ir;r2'N srs?co So ^ ^-5 « t CHAP, v.] TABLBS, &C., FOB WEIRS. 271 Look for the Telocity of water in the pipe in the upper rew, and io the colamn below it, and opposite to the given diameter of the pipe studing in the last column, will be found the perpendicular ha^ of a column or head, in feet, inches, and tenths, requisite to orercome the friction of such pipe for 100 feet in length, and obtain the giren Telocity. 22, TaUe containing the qttantity of Water discharged over an inch vertical section of a Weir. Depth of the upper edge of the waste- boaid below theiur&ce in English inches. Cnbic feet of water discharged in a minute by an inch of the waste- board, according to Du Buat's formuls. Cubic feet oi water discharged in a minute by an inch of the waste- board, according to experiments made in Scotland. Gallons of 282 inches, corresponding with results in col. 3. 1 0-403 0-428 2-621 2 1-UO 1-211 7-417 3 2-095 2-226 13-634 4 8-226 3-427 20-990 5 4-507 4-789 20-332 6 5-925 6-295 38-357 7 7-466 7-983 48-589 8 9-122 9-692 59-364 9 10-884 11-664 70-826 10 12-748 13-535 83-164 11 14-707 15-632 95-746 12 16-758 17-805 109-055 13 18-895 20-076 122-965 14 21-117 22-437 137-427 15 23-419 24-883 152-408 16 25-800 27-413 167-906 17 28-258 80024 183-897 18 30-786 82-710 200-350 ^^ To the aboTe table, originally due to Du Buaty is added a third ^^QiDD, containing the quantities of water discharged, as inferred J^Jffl experiments made in Scotland, and examined by Dr. Robison, Jo found that they in general gaTe a discharge -^ greater than that ^lich » deduced from Du BucSs formulas. We would recommend ^ therefore to the engineer to employ the third column in his prac- ^^ or the fourth if he wish for the result in gallons. . Ifdiej be odd quarters of an inch, look in the tabic for as many ^^<ie« as the depth contains quarters, and take the eighth part of ^inswer. Thus, for 3| inches, take the eighth part of 24-883, ^luch corresponds to 15 inches. This is 3*110. 23. The quantity discharged increases more rapidly than the width : ^ obUin a correct measure of it, if / be the width or length of tho 272 DI8CHAROX8 OVBB WEIBS. [PABT 11. waste-board in inches, take (i-^-^l) times the quantity for one inch of wasteboard of the given depth, from the preceding table. In the preceding table it is supposed that the water from which the discharge is made is perfectly stagnant; but if it should happen to reach the opening with any velocity, we have only to multiply the area of the section by the velocity of the stream. 24. When the quantity of water Q discharged over a weir, is known, the depth of the edge of the wasteboard, or H, may be approximated from the following formulae, / length of wasteboard. Q = 11-5/H^ nearly (XX.) or, more accurately by adding the correction in article 15, Q = ii-5(^ + 4) «* (^Xl) 26. The quantities discharged for any given width are as the ^ power of the depth, or as H *. Hence, to extend the use of the table to greater depths, wc have only for Twice any depth, take Q x 2*828 3 times Q x 5196 4 times Q x 8*000 5 times Q x 1M80 6 times Q x 14-697 7 times Q x 18*520 8 times Q x 22*627 9 times Q x 27000 10 times Q x 31-623 and the results will be nearly true. To make them still more correct, where great accuracy is required, add to them their thotisandth part. Easamples of the use of* the Tables and Rules. Ex, 1. Let the depth of a weir be 10 inches below the upper surface of the water, and the width 8 inches. How many cubic feet of water will be discharged in a minute ? cub.fiB«t By table § 23. Q for 10 inches deep and 1 inch wide = 13*535 Multiply this by / == 8 106-280 Add ^>jy of this product 5*314 Discharge in one minute = 111*594 CHAP. V.j STRBAli liSASUKSBt. 97d Em. S. Let the depth be 9 feet, and the width 1 fooL Required the cubic feet discharged in a minute. Bj the table Q for 12 inches deep and 1 inch wide s 17*805 Faetor for 9 times depth ss 27 Quantity for 1 inch width 480*735 Multiply by / » 12 5768-802 Add ^9 of the product 266*441 Total quantity in cubic feet s= 6055*261 Ex, 3. Let a square orifice of 6 inches each side be placed in a sluice-gate with its top 4 feet below the upper surface of the water : how much will it discharge in a minute ? Here the quantity discharged by a slit in depth 48 inches, must be taken from one in depth 54 inches. cubic feet. For 54, multiply Q for 6 by 3' or 27 169*965 For 48, Q for 12 by 4^ or 8 142440 Difference 27*525 S7*525 X (6 + ^) as 173*4 cubic feet, quantity discharged. i^(C^.<— In an example like this, it is a good approximation, to 9nuUiyUf confinuaify together the area of the orifice^ the number 336*, €^n£l ike equare root of the depth infect of the middle of the orifice. Thus, in the preceding example, it will be | x ^ x 336 x >/4*25 3= ^ X 336 X 2*062 = 173*2 cubic feet. The less the height of the orifice compared with its depth under ^e water, the nearer will the result thus obtained approach to tte truth. If the height of the orifice be such as to require consideration, the 2Mnciple off 7, page 261, may be blended with this rule. Thnsy 9km\jmg ^is rule to Ex. 2, we shall have area x s/ depth ^ dd6 X I ss 9 X 3 X 224 = 6048, for the cubic feet discharged. ^niis is less than the former result by about its 900th part. It is, ttlierefore, a good approximation, considering its simplicity: it may in lEKianj VMes supersede the necessity of recurrence to tables. Skcy. III. Coniriveaices to measure the velocity of running waters. 1. For these purposes, various contrivances have been proposed, of ^hieh two or three may be here described. 836 « 6 6 x 60. 274 STREAM MBASCREBS. [PART II. Suppose it be the Telocity of the water of a river that is required ; or, indeed, both the Telocity and the quantity which flows down it in a given time. ObserTe a place where the banks of the nTer are steep and nearly parallel, so as to make a kind of trough for the water to run through, and by taking the depth at Tarious places in crossing make a true section of the nTer. Stretch a string at right angles OTer it, and at a small distance another, parallel to the first Then take an apple, an orange, or other small ball, just so much lighter than water as to swim in it, or a pint or quart bottle partly filled with water, and throw it into the water aboTO the strings. ObserTe when it comes under the first string, by means of a quarter- second pendulum, a stop watch, or any other proper instrument; and obsenre likewise when it arrives at the second string. By this means the Telocity of the upper surface, which in practice may frequently be taken for that of the whole, will be obtained. And the section of the riTer at the second string must be ascer- tained by taking various depths, as before. If this section be the same as the former, it may be taken for the mean section: if not, add both together, and take half the sum for the mean section. Then the area of the mean section in square feet being muldplied by the distance between the strings in feet, will give the contents of the ^'ater in solid feet which passed from one string to the other during the time of observation : and this, by the rule of three, may be adapted to any other portion of time. The operation may often be greatly abridged by taking notice of the arrival of the float- ing body opposite two stations on the shore, especially when it is not convenient to stretch a string across. An arch of a bridge is a good station for an experiment of this kind, because it affords a very regular section and two fixed points of observation ; and in some instances the sea practice of heaving the log may be advantageous. Where a timepiece is not at hand, the observer may easily construct a quarter- second or other pendulum, by means of the rules and table relating to pendulums in the Dynamics. 2. M. Pitot invented a stream measurer of a simple construc- tion, by means of which the velocity of any part of a stream may readily be found. This instrument is composed of two long tabes of"" glass, open at both ends; one of these tubes is cylindrical throughout; z the other has one of its extremities bent into nearly a right angle, ^ and gradually enlarges like a funnel, or the mouth of a trumpet:^ these tubes are both fixed in grooTes in a triangular prism of wood,^ so that their lower extremities are both on the same level, standings thus one by the side of the other, and tolerably well preserred from^ accidents. The frame in which these tubes stand is graduated, closed by the side of them, into dirisions of inches and lines. To use this instrument, plunge it perpendicularly into the watery in such manner that the opening of the funnel at the bottom of ontfS of the tubes shall be completely opposed to the direction of the cur— - rent, and the water pass freely through the funnel up into the tube-i^ Then observe to what height the water rises in each tube, and note^ CHAP, v.] 8TRBAM MBA8UREB8. 275 the difference of the sides, for this difference will be the height due to the velocity of the stream. It is manifest, that the water in the cylindrical tuhe will be raised to the same height as the surface of the stream by the hydrostatic pressure, while the water entering from the current by the funnel into the other tube will be compelled to rise aboTe that surface by a space at which it will be sustained by the impulse of the moring fluid: that is, the momentum of the stream iriU be in equilibrio with the column of water sustained in one tube fhore the surface of that in the other. In estimating the velocity by means of this instrument, we must have recourse to theory, as cor- rected by experiments. Thus, if A, the height of the column sus- tained by the stream, or the difference of heights in the two tubes, be in feet, we shall have e = 6'5 v^ A, nearly, the velocity, per second, of the stream; if A be in inches, then o = 22*47 %/ ^i nearly: or &rther experiments made with the same instrument may a little modify these coefficients. It win be easy to put the funnel into the most rapid part of the stream, if it be moved about to different places until the difference of altitade in the two tubes becomes the greatest. In some cases it will happen, that the immersion of the instrument will produce a little eddy in the water, and thus disturb the accuracy of the observ- «tioii : but keeping the instrument immersed only a few seconds will correct this. The wind would also affect the accuracy of the experiments ; it is, therefore, advisable to make them where there is little or no wind. By means of this instrument a great number of curious and useful observations may easily be made: the velo- city of wator at various depths in a canal or river may be found with tolerable accuracy, and a mean of the whole drawn, or they maj be applied to the correcting of the theory of waters running down gentle slopes. The observations may likewise be applied to nmeertBon whether the augmentations of the velocities are in propor- tion to the increase of water passing along the same canal, or what other relations subsist between them, &c. Where great accuracy is not required, the tube with the funnel %t hottom will alone be sufficient, as the surface of the water will be Indicated with tolerable precision bv that part of the prismatic fimme for the tube which has been moistened by the immersion : and the wdocUks may be marked against those altitudes in the tube which indicate them. d. Another good and simple method of measuring the velocity of water in a canal, river, &c., is that described by the Abbe Mann^ in Itis treaitise on rivers; it is this: — Take a cylindrical piece of dry light ^wood, and of a length something less than the depth of the water in the river ; about one end of it let there be suspended as many small weights as may keep the cylinder in a vertical or upright position, with ita head^just above water. To the centre of this end fix a •mall straight rod, precisely in the direction of the cylinder's axis, in order that, when the instrument is suspended in the water, the devia* T 9 276 STRSAM MKA8UREBS. [l tions of the rod from a perpendicularity to the mvrfmce of indicate which end of the cylinder goes foremost, by wbieh discovered the different velocities of the water at different for when the rod inclines forward, according to the direction current, it is a proof that the surface of the water has the velocity ; but when it reclines backward, it shows that the current is at the bottom ; and when it remains perpendiculai sign that the velocities at the top and bottom are equal. This instrument, being placed in the current of a river o receives all the percussions of the water throughout the whol and will have an equal velocity with that of the whole carre the surface to the bottom at the place where it is pat : by that means may be found, both with exactness and e mean velocity of that part of the river for any determinate and time. But, to obtain the mean velocity of the whole section of tl the instrument must be put successively both in the mid towards the sides, because the velocities at those places are of different from each other. Having by these means found the velocities, from the spaces run over in certain times, the aritl Tuean proportional of all these trials, which is found by divi< common sum of them all by the number of the trials, will mean velocity of the river or canal. And if this medium vel multiplied by the area of the transverse section of the waten place, the product will be the quantity running through that ; a second of time. The cylinder may be easily guided into that part which ^ to measure, by means of two threads or small cords, wh persons, one on each side of the canal or river, must hold and taking care at the same time neither to retard nor accelei motion of the instrument. 4. Let A A' B B' be a hollow cylinder, open at both ends, it be capable of being fixed by the side of a platform or of a that its lower extremity BB^ may be placed at any proposed depth below H R, the upper surface of the stream. Let P P^ be pulleys, fixed at opposite sides of the top and bottom of the tube. To O, a globe of specific gravity nearly the same as that of water, let a cord OP' PS be attached, passing freely over the pulleys P', P, and having sufficient length towards S to allow of its running off to any convenient distance. Then, the bottom of the tube being immersed to any proposed depth, let the globe G be exposed to the free operation stream; and as it is carried along with it, it will in 1, 2, i seconds, or any other interval of time, drew off from a fizei as S, a portion of cord; from which and the time elapsed, the at the assigned depth will become known. ] STBBAM MBA8URBB8. 277 mvented, in 1720, an instrument called the Marine 'y for the doable purpose of measuring a ship's way, and ing the velocities of streams. It is described in the Phil. oL dd ; and in the succeeding volume a curious example of given in ^ tables showing the strength and ffradual increase aw of the tides of flood and ebb in the nver Thames, as in Lambeth Reach." They are too extensive to be inserted are very interesting, and may be seen in the Philosophical ms AMdptedy vol. vii. p. 133. 378 EQUILIBRIUM OF ILASTIC FLUIDS. [PABT II. CHAP. VI. PNEUMATICS. Sect. I. Weight and Equilibrium of Air and EUutic Fluids. 1. The fundamental propositions that belong to hydrostatics are common to the compressible and incompressible fluids, and need not, therefore, be repeated here. 2. Atmospheric air is the best known of the elastic fluids, and has ^ been defined to be an elastic fluid, having weight, and resisting com*^ ^ pression with forces that are directly as its density, of inversely as ^ the spaces within which the same quantity of it is contained. The correctness of this definition is confirmed by experiment. 3. The weight of air is known from the Torricellian experiment, ^^^ or that of the barometer. The air presses on the orifice of the ^^. inverted tube with a force just equal to the weight of the column of ~% ^ mercury sustained in it. The weight of a cubic foot of air, with the barometer at 30 inches, ^ -r^ and the temperature ^5^ of Fahr., is 1*2 ounces avoirdupois, or-^^3 about v^rd part of the weight of a cubic foot of water under the — ^ i< same circumstances. A bottle, weighed when filled with air, is found heavier than after^r ^i the air is extracted. The mean pressure of the atmosphere at^ -^t London is 14*18 lbs. on every square inch of the earth's surface^p^^i which is equivalent to the pressure of a column of quicksilver 28*89^^^ ^ inches in height; or a column of water nearly 33 feet in height— =^^* Hence the total pressure on the convex surface of the earth amount^^ -^^ to 10,686,000,000 hundreds of millions of pounds. The elastic force of the air is proved, by simply inverting a vesseT ^^*] full of air in water : the resistance it offers to farther immersion, an^^ -•^ the height to which the water ascends within it, in proportioira^^ as it is farther immersed, are proofs of the elasticity of the aixi^'C'^ contained in it*. * It is in virtue of this property, and ought to be known as ottensivdy i possible, that a man*s hat will serve in most cases as a temnormry life-preMrva to persons in hazard of drowning, by attending to the following direcdoos :- When a person finds himself in, or about to be in, the water, let him lay hok^ ^^ of his hat between his hands, laying the crown dose under his chin, and th^^ -^ mouth under the water. By this means, the quantity of air conuined in th^' ^ cavity of the hat will be sufficient to keep the head above water for icvtii^^ ' hoars, or until assistance can be rendered. i CHAP. VI.] WBIOHT AND EQUILIBRIUM OP ELASTIC FLUIDS. 279 4. When air is confined in a bent tube, and loaded with diflferent weights of mercury, the spaces it is compressed into are found to be inversely as those weights. But those weights are the measures of the elasticity; therefore the elasticities are inversely as the spaces which the air occupies. The densities are also inversely as those spaces; therefore the elasticity of air is directly as its density. This law was first proved by Manotte's experiments. ' In all this, the temperature is supposed to remain unchanged.-^ These properties seem to be common to all elastic fluids. Air resists compression equally in all directions. No limit can be assigned to the space which a given quantity of air would occupy if all compression were removed. 5. In ascending from the surface of the earth, the density of the air necessarily diminishes : for each stratum of air is compressed only by the weight of those above it ; the upper strata are therefore less compressed, and of course less dense than those below them. 6. Supposing the same temperature to be difl^used through the atmosphere, if the heights from the surface be taken increasing in arithmetical progression, the densities of the strata of air will decrease in geometrical progression. Also, since the densities are as the com- pressing forces, that is, as the columns of mercury in the barometer, the heishts from the surface being taken in arithmetical progression, the columns of mercury in the barometer at those heignts will decrease in geometrical progression. 7. Logarithms have, relatively to the numbers which they re- present, the same property, therefore if b be the column of mer- cury in the barometer at the surface, and at any height h above the surface, taking m for a constant coefficient, to be determined by experiment, A = in(log6 — logg), or A = mlog- (I.) where m may be determined by finding trigonoroetrically the value of A in any case, where b and $ have been already ascertained. 8. If ^ be the height of the mercury in the barometer at the lowest station, fi at the highest, t and f the temperatures of the air at those stations, f the fixed temperature at which no correction is required for the temperature of the air; and if g and / be the temperatures of the quicksilver in the two barometers, and n the ex- pansion of a column of quicksilver, of which the length is 1, for each degree of heat; k being the perpendicular height (in fathoms) of the one station above the other, il = 10000(1 + -00244/^'-^ -A log —^ 77^] (II.) « being nearly = j^. 280 ATMOSPHBRIO ALTlTUDli. [PABT 11. If the oentigrade thermometer it used, hecaase the hegtHning of the scale agrees with the temperature f^ so that /*« 0» the formula hecomes more simple ; and if the expansion for air and nercurj he hoth adapted to the degrees of thitf scale, *= ,0000{. + 0044l(l±i:) log __J__^} (HI.) 9< The temperature of the air diminishes on ascending into the Ittmosphere, hoth on account of the greater distance from the earth, the principal source of its heat, and the greater power of aheorbing heat that air acquires by being less compressed. 10. Professor Leslie, in the notes on his EkmenU qf Geometry^ p. 495 (edit. 2nd), has given a formula for determining the tempera- ture of any stratum of air when the height of the mercury in the barometer is given. Tlie column of mercury at the lower of two stations being &, and at the upper /?, the diminution of heat, in degrees of the centigrade thermometer, is f - — T f 2^* Which seems to agree well with observation. 1 1 . If the atmosphere were reduced to a body of the same density which it has at the surface of the earth, and of the same temperature, the height to which this homogeneous atmosphere would extend is, in fathoms, equal to 4343 ^1 + '00441 ——V or, taking the expansion according to Laplace, = 4343 ( 1 + ttt^ ) • • • C^^*) Hence if h be the height of the mercury in the barometer, reduced to the temperature /, the specific gravity of mercury is to that of airi as ft to 4343 ( I + Tjwjj: J ^ or the specific gravity of air h 72 X 4343 X (\ ^ ~\ ' \ ^ 1000/ The divisor 72 is introduced in consequence of b being expressed in inches.— (jP/tfj^atV^ Outiines.) 12. The^ Telocity with which air rushes into a vacuum is equal to that which a heavy body would acquire in Mling from a height equal to that of a homogeneous atmosphere equivalent in wemit to that of the air at the time. Thus if H be the height of Uie homogeneous atmosphere, deduced from the formula (IV.) abote, expressed in feet, and V the velocity of the air in feet per second^ Wd have from the formula, page 215, V = S- <V.) CHAP. VI.] ATM08PHBBIC ALTITUDES. 281 From wbieh we find the value of V for the mean temperature and preasare to be 1860 feet per second. 13. The velocity with which sound traverses the atmosphere de- pends not only upon the direction and force of the wind, but also on the temperature and density of the air at the time, and the quantity of moisture which it contuns. In the following formula given by Mr. Galbraitb, of Edinburgh, the whole of these circumstances are taken into account Let v be the velocity of sound in feet per second, t the temperature of the air, 8 the elastic force of vapour for the due point, the barometric pressure, x the latitude of the place of observation, v the velocity of the wind, and ^ the angle which the direetion of the wind makes with that of the sound ; then »a= (102-4225 + 01103 0(l + ^i qZ. 2e ) C^^'^'''^® — 0-01378. cos 2 a) H- w. cos ^ (VI.) 8bct. II. Madtine$ for BaUing Water by the Pressure of the Atmoephere. 1. The term Pump is generally applied to a machine for raising water by means of the air's pressure. 2. The common suction-pump consists of two hollow cylinders, which have the same axis, and are joined in A C (fig. 216). The lower is partly immersed, perpendicularly, in a spring or reservoir, and is called the suction-^ube; the upper the body of the pump. At AC is a fixed sucker containing a valve which opens upwards, and is less than 34 feet from the surface of the water. In the body of the pump is a piston D made air-tight, movable by a rod and handle, and con- taining a valve opening upwards. And a spout G is placed at a greater or less distance, as convenience may require, above the greatest elevation of D. The action of this pump is as follows. Suppose the movable pis- ton D at its lowest depression, the cylinders free from water, and the air in its natural state. On raising this piston, the pressure of the air above it keeping its valve closed, the air in the lower cylinder A B forces open the valve at A C, and occupies a larger space, viz., be- tween B, the surface of the water, and D ; its elastic force, therefore, being diminished, and no longer able to sustain the pressure of the external air, this latter forces up a portion of the water into the cylin- der AB to restore the equilibrium. This continues till the piston has reached its greatest elevation, when the valve at A C closes. In its subeeqaent descent, the air below D becoming condensed, keeps the valve at AC closed, and escapes by forcing open that at D till the piston has reached its greatest depression. In the following turns a similar effect is produced, till at length the water rising in the cylin- der forces open the valve at A C, and enters the body of the pump ; 282 PUMPS. [part II. when, by the descent of D, the valve in AC is kept closed, and the water rises through that in D, which on re-ascending carries it for- ward, and throws it out at the spout O. 3. Cor, 1. The greatest height to which the water can be raised in the common pump by a single sucker is when the column is in eqoi- Hbrio with the weight of the atmosphere, that is, between 32 and 36 feet. 4. Cor. 2. The quantity of water discharged in a given time is de- termined by considering that at each stroke of the piston a quantity is discharged equal to a cylinder whose base is a section of the pump, and altitude the play of the piston. 5. To determine the force necessary to overcome the resistance experienced by the piston in ascending. Let h = the height H F (fig. 217) of the surface of the water in the body of the pump above EF, the level of the reservoir; and a^ = the area of the section MN. Let h I ^ the height of the column of water equivalent to the pres- sure of the atmosphere ; and suppose the piston in ascending to arrive at any position mn which corresponds to the height IF. It is evi- dent that the piston is acted upon downwards by the pressure of the atmosphere ^ d^ h.^ and by the pressure of the column B m = er X H I ; therefore the whole tendency of the piston to descend = a* (A + HI). But the piston is acted upon upwards by the pressure of the air on the external surface £ F of the reservoir = c^ hy\ part of which is destroyed by the weight of the column of water having for its base mn, and height FI; .*. the whole action upwards = a^ x (^j — FI); whence F = a* . (A, -f HI) = 0^.(^1— FI) = a«. FH == d'h, that is, the piston throughout its ascent is opposed by a force equal to the teeight of a column of water having the same hose as the piston^ and an altitude equal to that of the surface of the water in the bodjf of the pump above that in the reservoir. In order, therefore, to produce the upward motion of the piston, a force must be employed equal to that determined above, together with the weight of the piston and rod, and the resistance which the piston may experience in conse- quence of the friction against the inner surface of the tube ♦. * Suppose the body of the pump to be 6 inches in diameter, and the greatest height to which the water is raised to be 30 feet ; suppose, also, the weight of the niston and its rod to be 10 lbs., and the friction one-fifUi of the whole weight. Then, -^ of the square of the diameter gives the ale gallons in a yard in length of the cylinder, and an ale gallon weighs 10| Iba. Therefore (6< X 10) + % (6* X 10) »= 360 -h 7-4 " 367*4 lbs., weight of the opposing column of water. And 367 4 + 10 -f i (377*4) = 452*9 lbs., whole opposing pressure. If the piston rod be moved by a lever whose arms are as 10 to 1, this pres- snre will be balanced by a force of 46*29 lbs., and overcome by any greater force. CHAP. VI.] SUCTION PUMP. 383 When the piston begins to descend, it will descend by its own weight ; the onhr resistance it meets with being friction, and a slight impact against the water. 6. Cor. 1. If the water has not reached the piston, let its level be in e ir. The under surface of the piston will be pressed by the mtemal rarefied air. But this air, together with the oolumn of water, £ o, is in equilibrio with the pressure of the atmosphere a' A, ; and . -• its pressure =s a' . (Ai — E 9). And the pressure downwards = «•*.; Hence the force requisite to keep the piston in equilibrio increases as the water rises, and becomes constant and = a'A as soon as the water reaches the constant level B H. 7- Cor, 2. If the weight of the piston be taken into the account, let this weight be equal to that of a column of water whose base is m II and height/), ss a*/? ; .-. F = (i'. (Er+/?). 8. To determine the height to which the virater will rise after one stroke of the piston ; the fixed valve being placed at the junction of the suction-tube and body of the pump : supposing that afler eyery elevation of the piston there is an equilibrium between the pressure of the atmosphere on the surface of the water in the reservoir, and the elastic force of the rarefied air between the piston and surface of the column of water in the tube, together with the weight of that column. Let ah {j^g, 218) be the surface of the water in the suction-tube, after the first stroke of the piston : if the piston were for an instant stationary at D, the pressure of the atmosphere would balance E 6, and the elastic force of the air in N a. Let A E, the height of the suction-tube, = a, D R, the play of the piston, == &, h s: the height of a column of water equivalent to the pressure of the atmosphere, y ^ the height of a column equivalent to the pressure of the air in No, m =E Eo, ^ =r 3*14159 and B and r = the radii of the body and the suction tube. Then « + jr = A, AF vHa hf^a I hr'* a whence A = ^0 + ^- , . , -r- ; .-. AR«6 + hf*a — hr*a = R**« + r*a« — r*a?» -f- ht^a. -*y^* ^ reduced vj^^^li^^rf^^ ^.«. these arc ^^ ^^^^^^^^Tii^^^^. ^-^ — - — TnX. «A BO on. ,^ " 1 ^ »8ceot8' ?fc+« —*•--" ' rise aftet^'/tVve «'*-*'" CHAF. VI.] SUCTION PUMP. 285 Uon due to each particular stroke, the differences of those elevations, and the successive differences in the elastic force of the remaining air, may be known. 10. If the weight of the valve c be not considered, it is evident that after a certain number of strokes a vacuum will be produced in the suction-tube, provided it be equal to or not greater than the height due to the pressure of the atmosphere, that is, if a be not greater than h. For, in this case, x^ =. x^.,, •nd.-.«^,= J{jt,— y^T* — 4Am*-4a?. . . (A -f o — «.-i)}, (IX.) whence «»., = ky the greatest height of the column of water in the tube. If, therefore, the length of the suction-tube do not exceed the height due to the pressure of the atmosphere, the water will continue to ascend in it after every stroke of the piston, till at length it will into the body of the pump. But if the altitude of A F be greater than k, the water will con- tinue to ascend without ever reaching its maximum height. For, in this case, an actual vacuum cannot be produced ; and as ^^ -f y* = ^'* and y, can never become = 0; .*. ^« can never = ^*. But, tlie successive values of ^r continually decreasing, the corresponding values of as will continually increase. 11. If the weight of the valve c be taken into the account, a column of water must be added equal to the additional pressure to be overcome. Let / = the height of this column, then a + y^l^h; and .'. a + y ^ h ^ I = hy If therefore this value of A| be substituted for A, the preceding equations are applicable. 12. In the preceding cases, the movable piston has been supposed to descend to A C. If it does not, it may happen that the water may not reach AC, though AC be less than' 34 feet from the surface of the water in the reservoir. After the first elevation of the movable piston to its greatest alti- tude, c being closed, the elastic force of the air between DN and AC is (A — «), and its magnitude x h R*. If, in descending, the piston describes a space ^ less than ^, so as to stop at a distance b ^ b* from AC, this magnitude becomes {b -- b") . v B,^ ; .*. the elastic force is (A — «) . Y 7T. Now in order that the pressure upwards may open the valve, this must exceed the elastic force of the atmosphere ; * Hence it appears that it is not ttrieUy true, that wat«r will ascend In the •action-tube to a height equal that of a column equivalent to the pressure of the atmoephere. This is a limit to which it approximates, but does not reach in a finite time. 286 SUCTION PUMP. [part II, SUCTION PUHP. .-.(*-«), h or (A - *) . i >h.{h-b'); .:hx<Lhh', X V If .% T-be less than -, the valve DN will Dot open; there h will therefore be the same quantity of air between A C and the sucker: which, when the piston has reached its highest elevation, will have the same elastic force as that between A C and a' h' ; and therefore c, being equally pressed on both sides, will remain un- moved, and the water will not ascend. 13. If the fixed valve be placed at the surface of the water; to determine the ascent of the water in the suction-tube. Let £ a, £ a^ be the successive heights to which the water rises ; then, after the first ascent of the piston, mo -f a — « whence a? =: J . {/? — -Jp^ — ^hmh} and y = i . {2k — p -f >//?* — 4tkmh}^ which equations are the same as were determined for the first ascent of the piston (§ 8). Therefore, in the same manner as before, we shall have «, = J . {/? — s/p^ — 4>hmh — 4Aa?,.,}, 14. If the water be supposed to stop after (n -f IX ascents of the piston, then «, = a?»_, ; and .*. d?,_, = i . {/> — "^p^ — ^hmh — 4Aa?^,}, whence d?,_i = J . {a -f mh ± >/(a + mhY — ihmb} ...(X.) Hence, therefore, there are two altitudes at which the water may stop in its ascent, if (a + mbY is equal to or greater than 4 Am 5. In the former case the two values of x^^^ are equal, that is, there will be only one altitude = ^ . (a + m 6), at which the water will stop. In the latter case there are two which may be ascertained. If 4 Am* be greater than (a + nthf^ the water will not atop. Ex.l. If A = 32 feet, a = 20, 6 = 4, and m = 1, or the suction-tube and body of the pump be of the same diameter. «— I = I . {20 + 4 ± V(24)« - 4 . 1 . 32 . 4} = i . {24 ± V64} = le or 8. CHAP. VI.] VORCINO PUMP. 287 Ex. 2. If A = 32 feet, a = 25, ft = 2, and wi = 4, x^. = 1 . {25 H- 8 ± >/(33)' — 4 . 32 . 4 . 2} = J . {33 ± >/65} = 41-8062 or 241938. 15. If m = 1, or the tubes have the same diameter, «-. = i . {« + ft ± >/(« + hf - 4A6}, which is imaginary, if (a + 5)^ is less than 4 A ft, or ft greater than (o + fty 4A ' In order, therefore, that this pnmp may produce its effect, the play of the piston muit be greater than the sqtiare of its greatest akitttde above Vie surface of the water in the reservoir divided by four times the height due to the pressure of the atmosphere, 16. The lifing-pump consists of a hollow cylinder, the body of which is immersed in the reservoir. It is furnished with a movable piston, which, entering below, lifts the water up, and is movable by means of a frame which is made to ascend and descend by a handle. The piston is furnished with a valve opening upwards. A little below the surface of the water is a fixed sucker with a valve opening upwards. This is an inconvenient construction, upon the peculiari- ties of which we need not dwell. 17- The forein^'pump consists of a suction-tube AEFC (fig. 219) partly immersed m the reservoir, of the body of the pump ABOC, and of the ascending tube HK. The body is furnished with a movable solid sucker or plunger, D, made air-tight. And at AC and U are fixed suckers with valves opening upwardd. 18. To explain the action of this pnmp. Suppose the plunger D at its greatest depression ; the valves closed, and the air in its natural state. Upon the ascent of D, the air in A CD occupying a greater space, its elasticity will be diminished, and consequently the greater elasticity of the air in A F will open the valve at A C, whilst that at H is kept closed by the elasticity of the external air ; water there- fore will rise In the suction -tube. On the descent of D from its greatest elevation, the elasticity of the air in the body of the pump will keep the valve A C closed, and open that at H, whence air will escape. By subsequent ascents of the piston, the air will be ex- pelled, and water rise into the body. The descending piston will then press the water through the valve at H, which will close, and prevent its return into the bodv of the pump ; D therefore ascending again, the sp^ce left void will be filled by water pressing through the valve AC; and this upon the next ascent of D is forced into the ascending tube ; and thus, by the ascents and descents of D, water may be raised to the required height. 19. In this pump D must not ascend higher than about 32 feet from the surface of the water in the reservoir. 20. To determme the force necessary to overcome the resistance experienced by the piston : — 288 FIRE INOINB. [past II. Let h = the height of a column of water equivalent to the presiare of the atmosphere, and E B the height to which the water is forced. Let M N he any position of the piston D whose area = A, and the weight of the piston and its appendages = P. Let X = the force necessary to push the piston upwards during the suction, friction not heing considered, and V s= that employed to force it down. When the piston ascends, and H is closed X = P + AA — A.(A — ME) = P -f A . ME. Let the sucker he in the same position in its descent, and therefore A C closed, and H open, Y = AA -fA.MB — (AA -f P) = A . M B — P. Hence X + Y = A.EB; or the whole force exerted, in the case of equilihrium, is equal to the weight of a column of water whose base is equal to that of the piston, and altitude the distance between the surface of the water and the point to which it is to be raised. 21. In this pump the effort is divided into two parts, one opposed to the suction, and the other to the forcing ; whereby an advantage is gained over the other pumps where the whole force is exerted at once whilst the water is raised. 22. In order to have the force applied uniform, lot X = Y ; .-. P-|-A.ME==A.MB — P; .-. P = JA . (MB — ME). The piston therefore must play in such a manner that M B may be - greater than M E. 23. In the common forcing pump, the stream b intermitting ; for ' there is no force impelling it during the return of the sucker. One mode of remedying this, is by making an interruption in the ascending tube, which is surrounded by an air vessel T (fig. 220); in which, when the water has risen above Z, the air above it is com- pcssed, and by its elasticity forces the water up through Z; the orifice of which is narrower than that of the tube, and therefore the quantity of water introduced during the descent of the piston will supply its discharge for the whole time of the stroke, producing a continued stream. 24. The fire-engine consists of a large receiver A B C D, called the air-vessel, into which water is driven by two forcing-pumps EF, OH (fig. 221), (whose pistons are Q an^ R), communicating with its lower extremities at I and K, through two valves opening inii-ards. From the receiver proceeds a tube M L through which the water is thrown, and directed to any point by means of a pipe movable about the extremity L. The pumps are worked by a lever, so that whilst one piston descends the other ascendi. The pumps communicate with a reservoir of water at N. CHAP. VI.] QUICKSILVER PUMP. 289 25. To explain the action of this engine. The tube N being immersed iu the reservoir, and the piston R drawn up, the pump OH becomes filled; and the descent of the piston R will, as in the forcing-pump (§ 1 8), keep the valve H close, and cause the water to pass into the air-vessel by the valve I, whilst, by the weight of the water in the air-vessel, the valve K will be kept abut. In the same manner, when R ascends, Q descending will force the water through K into the air-vessel. By this means the air above the surface of the water becoming greatly compresse<1 will, by its elasticity, force the water to ascend through M L, and to issue with a great velocity from the pipe*. 26, When the air-vessel is half full of water, the air being then compressed into half its natural space will have an elastic force equal to twice the pressure of the atmosphere : therefore, when the stop-oock is turned, the air within pressing on the subjacent water with twice the force of the external air, will cause the water to spout from the engine to the height of (2 — 1)83, or 33 feet; except so far as it is diminished by friction. Or, genemlly, if denote the fractional height of the water in II the air-barrel, then - will denote the height of the space occupied by the compressed air, n times the pressure of the atmosphere its elastic force, and (» — 1 ) 33, the height in feet to which the water may be projected. Thus, if I of the air-barrel be the height of the water, the elastic force of the air will be four times the pressure of the atmosphere, and (4 — 1) 33 =s 99 feet, the height to which the water may then be thrown by the engine. 27- The modifications in the constructions of pumps with a view to their practical applications are very numerous. Those who wish to acquaint themselves with some of the most useful, may consult the 2nd Tolume of my Treatise on MechanicSy and Nos. 13, 41, 69, and 93, of the Meehaniee Magazine, Id addition to these, there may now be presented a short account of uauieksiiver-vump^ which has been recently invented by Mr. Thomas Clark of Edmburgh, and which works almost without friction. It has great power in dra\nng and forcing water to any height, and is extremely simple in its construction. In fig. 222, a a is the main pipe inserted into the well b ; a valve is situated at c, and another at dj both opening upwards ; a piece of iron tube is then bent into a drcular form, as at/^ again turned off at g in an angular direction, so as to pass through a stuffing box at A, and from thence bent out- wards as at t, connecting itself with the ring. A quantity of quick- * Tbe preceding part of this section Is taken from Bland*s Hydrottatics ; a nxj dqgant and valuable work, which I beg most cordially to recommend to those who wish to obtain a oomprehenaive knowledge of the Uieory of this -■ ■ — at of adeaee. 290 SPIRAL PUMP. [part II. silver is then put into tlie ring filling it from q to q\ and the ring being made to vibrate upon its axis hy a vacuum is soon effected in the main pipe by the recession of the mercury from p to q^ thereby causing the water to rise and fill the vacuum : upon the motion being reversed, the quicksilver slides back to ^, forces up the water and expels it at the spout e. *' Mr. Clark calculates that a pump of this description with a ring twelve feet in diameter, Hdll raise water the same height as the common lifting pump, and force it one hundred and fifty feet higher without any friction." {Mechanics' Register and Jamiesons Edin- burgh Journal,) 28. It is usual to class with pumps, the machine known by the s^ name o^ Archimedes* scretCy or the teater-stiail. This consists either -im-t of a pipe wound spirally round a cylinder, or of one or more spiral J'.m^ excavations formed by means of spiral projections from an internal X^^a^ cylinder, covered by an external cylindrical case, so as to be wator-^- -r3cr tight. The cylinder which carries the spiral is placed aslant, so as io^^M ^\o be inclined to the horizon in an angle of from 30^ to 45°, and£» M-nd capable of turning upon pivots in the direction of its axis posited atirwEx at each extremity. The lower end of the spiral canal being immersedC» -^^ed in the river or reservoir from which water is to be raised, th^ MrMiie water descends at first in the said canal solely by its gravity; but «l.» *ut the cylinder being turned, by human or other energy, the water '^^ -^er moves on in the canal, and at length it issues at the upper extremitj^-^ ^^7 of the tube. Several circumstances tend to make this instrument imperfect ana» ^rsi^ inefficacious in its operation. The adjustments necessary to insure ^* ^ a maximum of effectual work are often difficult to accomplish. 1 ^^ " seldom happens, therefore, that the measure of the work done ex— :^^" ceeds a tliird of the power employed : so that this apparatus, not— -^^'* withstanding its apjmrent ingenuity and simplicity, is very sj>aringljC— -^ introduced by our civil engineers. 29. Spiral pump. This machine is formed by a spiral pipe or ^' several convolutions, arranged either in a single plane, as in fig. 223. > or upon a cylindrical or conical surface, and revolving round an axis ^ * The curved pipe is connected at its inner end, by a central water tight joint, to an ascending pipe, r P, while the other end, S, receivec. .^ during each revolution, nearly equal quantities of air and water««i^ This apparatus is usually called the Zurich machine, because it wa^ invented, about 1746, by Andrew Wirtz^ an inhabitant of Zurich*^ It has been employed with great success at Florence, and in Russia^ and the late Dr. Thomas Young states, that he employed it advantage— ously for raising water to a height of forty feet. The outer end of the pipe is furnished with a spoon, S, which contains as much water as will half fill one of its coils. The water enters the pipe a little before the spoon has reached its highest position, the other half remaining full of air. This air communicates the pressure of the column of water to the preceding portion; and in this manner the effect of nearly all the water in the wheel is united, and becomes capable of CHAP. VI.] SCHEMNITZ VESSELS. 291 sapporting tlie column of water, or of water mixed with air, in the ascending pipe. The air nearest the joint is compressed into a space much smaller than that which it occapied at its entrance ; so that, when the height is considerahle, it becomes advisable to admit a larger portion of air than would naturally fill half the coil. This, while it lessens the quantity of water raised, lessens also the force requisite to turn the machine. The loss of power, supposing the machine well constructed, arises only from the friction of the water against the side of the pipes, and that of the wheel on its axis : and where a large quantity is to be raised to a moderate height, both of these sources of resistance may be rendered very inconsiderable. 30. Schemnitz vessels^ or the Hungarian machine. The media- tion of a portion of air is employed for raising water, not only in the spird pump, but also in the air-vessels of Schemnitz, as shown in fig. 224. A column of water, descending through a pipe, C, into a closed reservoir, B, containing air, obliges the air to act, by means of a pipe, D, leading from the upper part of the air-vessel, or reservoir, on the water in a second reservoir. A, at any distance either above it or below it, and forces this water to ascend through a third pipe, E, to auj height less than that of the first column. The air-vessel is then emptied, the second reservoir filled, and the whole operation repeated. The air, however, must acquire a density equivalent to tlie requisite pressure before it can begin to act : so that, if the height of the columns were thirty-four feet, it must be reduced to half its natural space before any water could be raised, and thus half of the force would be lost. But where the height is small, the height lost in this manner is not greater than what is usually spent in overcoming friction, and other imperfections of the machinery employed. The force of the tide, or of a river rising and falling with the tide, might easily be applied to the purpose of raising water by a machine of this kind. Thus, if at low tide the vessel A were filled with air, then, at high tide, the water flowing down the tube £, would cause the water in the vessel B to ascend in the pipe C. 31. The hydraulic ram. In this hydraulic arrangement, the momeiitum of a stream of water flowing through a lon^ pipe is employed to raise a small quantity of water to a considerable heiglit. The passage of the pipe being stopped by a valve which is raised by the stream, as soon as its motion becomes sufiiciently rapid, the whole column of fluid must necessarily concentrate its action almost instantaneously upon the valve. In these circumstances it may be regarded as losing the characteristic property of hydraulic pressure, and to act almost as though it were a single solid : so that, supposing the pipe to be perfectly elastic and inextensible, the impulse may overcome almost any pressure that may be opposed to it. If another valve opens into a pipe leading to an air-vessel, a certain quantity of the water will be forced in, so as to condense the air, more or less rapidly, to the degree that may be required for raising a portion of the water contuned in it to a given height. The late Mr. Whitehurst appears to have been the first who u 2 292 FORCE OP THR WIND. [PART II. employed this method: it was afterwards improved by Mr. Boulton. But, like many English inyentione, it never was adequately estimated, until it was brought into public notice by a Frenchman. M. Mont- golfier, its re-inventor, gave to it the name wliich it now bears of the Hjfdravlic Ram, in allusion to the battering ram. The essential parts of this machine are represented in figure 225. When the water in the pipe A B (moving in the direction of the arrows) has acquired sufficient velocity, it raises the valve B, which immediately sto])s its farther passage. The momentum which the water has acquired then forces a portion of it through the valve, C, into the air-vessel, D. The condensed air in the upper part o( D causes the water to rise into the pipe £, as long as the effect of the <^^ ^e horizontal column continues. When the water becomes quiescent, ^..^.t the valve B will open again by its own weight, and the current along ^^ «-i| A B will be renewed, until it acquires force enougli to shut the said£» M^i valve B, open C, and repeat the operation. The motion in the horizontal tube arises from the acceleration oft^ <:^ o the velocity of a liquid mass falling down another tube, and com^ .fliiKim municating with this. In an experiment made upon an hydraulic ram, at Avilly, n eii- m^ -^jak Sen lis, by M. Turquet, bleacher, the expense of power was found t*,:^ U be to the produce, as 100 to 62. In anotlier, as 100 to 55; in tvr^-^^^wi others, as 1 00 to 57. So that a hydraulic ram placed in favourabi f ^Jblc circumstances, may be reckoned to employ usefully rather more thi mwM^Mn half its force. *i^* For more full accounts of the three last contrivanoes, tlr:^.JRlie reader may consuH the 2nd volume of my Mechanics, Sect. III. Force of the Wind. 1 . Air, when in continuous motion in one direction, becomes ^^ss a very useful agent of machinery, of greater or less energy, accordir .^iog to the velocity with which it moves. Were it not for its vsrialMlfir Mlitj in direction and force, and the consequent fluctuations in its auppT ^ "^ly? scarcely any more appropriate fii'st mover could generally be wnijl^ ^""^ for. And even with all its irregularity, it is still so useful as to require a separate consideration. 2. The force with which air strikes against a moving surfaoe, with which the wind strikes against a quiescent surface, is nearly the square of the velocity : or, more correctly, the exponent of i velocity, determined according to the rule siven in Example page 96, varies between 2*03 and 2*05; so Uiat, inmost practical/ cases, the exponent 2, or that of the square, may be employed wiC^ out fear of error. U $ he the angle of incidence, s the surface struct in square feet, and v the velocity of the wind, in feet, par teoond; then, if / equals the force in avoirdupois pounds, either of die two following approximations may be used, viz. : — CHAP. VI.] FORCB OP THE WIND. 293 /= 440 (XI.) or,/= -002288 c'^wn'jg (XIT.) Of these, the first is usually the easiest in operation, requiring only two lines of short division, viz., by 40 and by 1 1 . If the incidence be perpendicular, sin^/3 = 1, and these become, /= ^ = -002288 «»« (XIII.) 3. The table in the margin exhibits the Force of the wind when blowing perpen- dicularly upon a sur^Mse of one foot square, at the several velocities stated. The velo- city of 80 miles per hour, is that by which the aeronaut Gamerin was carried in his tiaUoon from Ranelagh to Colchester, in June, 1802. It was a strong and boister- ous wind, but did not assume the character of a hurricane^ although a wind with that velocity is so characterized in Rouse's table. In Mr. Green's aerial voyage from Leeds, in September, 1823, he travelled 43 miles in 18 minutes, although his balloon rose to the height of more than 4000 yards. 4. Borda found by experiment, in the year 1762, that the force of the wind is very nearly as the square of the velocity, but he iSBUDed that force to be ereater than what Rouse found (as expressed in die above tahle) in tne ratio of 111 to 100. Borda ascertained also^ as was natulfal to expect, that, upon difierent surfaces with the same velocity, the force increased more rapidly than the surface. M. Valz, applying ttie method of the minimum squai^es to Borda's resoltSy ascertained tiiat the whole might be represented by the formula ^ = 0001289 a^ + 0000030541 ar* (XIV.) and nearly as correctly by y = 0-00108 »" (XV.) ^ representing the surface in square inches (French), and p the force oorrespondibg to the velocity of 10 feet per second expressed in French pounds* Velocity of the Wfnd. Perpendi- cular force Milen = feet foot in fa one inane •▼OlIttupOlB pound*. hour. 1 second. 1-47 •005 8 2-93 •020 3 4-40 •044 4 5-87 •079 5 7-33 •123 10 1407 •402 15 2200 I 107 20 20*34 1-908 25 30*67 3075 30 4401 4-420 35 51 34 0027 40 58-68 7-873 45 00 01 0063 60 7336 12-300 00 88 02 17-716 80 117-30 31-400 100 14670 40-200 294 WAT£R AS A MECHANICAL AOBNT. [PAHT ^j^ CHAP. VII. MECHANICAL AGENTS. Sect. I. Water as a Mechanical Anient, 1. The impulse of a current of water, and sometimes its weight and impulse jointly, are applied to give motion to mills for grinding corn and for various other purposes. Sometimes the impulse is ap- plied ohiiquely to float-boards in a manner which may be compre- hended at once by reference to a amoke-Jack^ in which the asoendifiS smoke strikes the vanes obliquely, and communicates a rotatory in*" tion. If we imagine the wLole mechanism to be inverted, m^'^*^ to fall upon the vanes, rotation would evidently be produced ; m^^, that with greater or less energy in proportion to the qoantitj ^ water and the height from which it falls. i Water-wheels of this kind give motion to mills in Germany, a- ^^ ^ some other parts of the Continent of Europe. I have also w""'^; mills of the same construction in Balta, the northernmost Sbetla^^^ isle. But wherever they are to be found, they indicate a very vT^^ perfect acquaintance with practical mechanics; as they occasion considerable loss of power. 2. Water frequently gives motion to mills, by means of what technically denominated an undershot wheel. This has a number ^ planes disposed round its circumference, nearlv in the direction m ^ its radii, and these float-hoards (as they are called) dipping into tl*^" stream, are carried round by it, as shown in fig. 226. The axle c^ the wheel, of course, by "the intervention of proper wheels an' ^^ pinions, turns the machinery intended to be moved. Where tb ^^ stream is large and unconfined, the pressure on each float-board is thi^^ corresponding to the head due to the relative velocities (or differenc^^^ between the velocities of stream and float-board) : this pressare v^^ therefore, a maximum when the wheel is at rest; but the wwrkfer-^'^ formed is then nothing. On the other hand, the pressure is noibin^^ when the velocity of the wheel equals that of the stream. Goose — "^ quently, there is a certain intermediate velocity, which CMises tb^^ work performed to be a maximum. The weight equal to the pressure is Q (^/ H — ^ ^)', Q beinp the quantity of water passing in a second, H the height due to V the velocity of the water, and h that due to U the velocity of the floefr- board. Considering this as a mass attached to the wheel, its moTing force is obtained by multiplying it into U : and as >/ H — ^^ A vmries CHAP. VII.] WATBBMJLLS. 295 as V — U, this moviDg force varies as (V — U)^ . U which is a maxi" mum when U = ^l V. In this case, then, the rim of the wheel moves with ^ of the velocity of the stream ; and the effect which it pro- duces is Q X (fV)« X 4V = ^QV»: 80 that the work performed hy an undershot wheel, according to the usual theory, equals -^-j of the moving force. Friction, and the resistance of fluids, modify these results; hut Smeaton and others have found that the maximum work is always obtained when U is between ^ V and ^ V. 3. Where the floats are not totally immersed, the water is heaped upon them ; and in this case the pressure is that due to 2 H. 4. When the float-boards move in a circular sweep close fitted to them, or, in general, when the stream cannot escape >vithout ac- quiring the same velocity as the wheel, the circumstances on which the investigation turns become analogous to what happens in the collision of non-elastic bodies. The stream has the velocity V before the stroke which is reduced to U, and the quantity of motion corre- sponding to the difference, or to V — U, is transferred to the wheel ; this turns with the velocity U; and therefore the eflfcct of the wheel (V— U\ VU— U* ,.,. I U, or — ; which is a maximum when V ^ 2 U ; being then ^ of the moving power. Hence appears the utility of constraining the water to move in a narrow channel. 5. The undershot wheel is used where a large quantity of water can be obtained with a moderate fall. But where the fall is con- siderable the overshot (fig. 227) is almost always employed. Its cir- cumference is formed into angular buckets, into which the water is delivered eithef at the top or within 60** of it : 52** | is the most advantageous distance. In that case, if r = the full radius of the wheel, H the whole, and h the effective height of the fall, A = r (1 -i- sin 37*"^) = 1-605 r, and r = '623 h. If the friction be aboat f of the moving power, the velocity of the circumference of the wheel to produce a maximum effect, will = 2*07 >/ H. Here, too, a fall of -^ H will give the water its due velocity of impact upon the wheel: and 122*176 s U^ equals the mechanical effect in pmrndfl, 8 being the section, in feet, of the stream that supplies the backets. Mr. Smeaton's experiments led him to conclude that overshot wheels do most work when their circumferences move at the rate of 3 feet in a second, and that when they move considerably slower than this, they become unsteady and irregular in their motion. This de- termination is, however, to be understood with some latitude. He mentions a wheel 24 feet in diameter, that seemed to produce nearly ils fall effect though the circumference moved at the rnte of G feet in a second ; and another of the diameter of 33 feet, of which the cir^ 296 WATEBMILLS. [PABT H. curafereiicc had only a velocity of 2 feet in a second, without any considerable loss of power. The first wheel turned round in 12*6 seconds, the latter in 51*9 seconds. 0. Where the fall is too small for an overshot wheel, it is most advisable to employ a breast^wheel (Plate VI.) which partakes of iu {)roperties ; its float-boards meeting at an ansle, so as to be assimi- ated to buckets, and the water beins: considerably confined within them by means of an arched channel fitting moderately close, but not so as to produce unnecessary friction. But when the circum- — stances do not admit of a breast- wheel, then recourse must be had to^:=a the undershot. For such a wheel it is best that the float-boards be^^ so placed as to be perpendicular to the surface of the water at th^^ j time they rise out of it ; that only one half of each should ever b»^^:3 below the surface, and that from 3 to 5 should be immersed at onc^»«^: The Abbe Mann proposed that there should not bo more than six c^ eight float-boards on the whole circumference. 7. Mills moved by the re-action of water are usually denominate^K^ Barkers Mills; sometimes, however. Parent's; at others, Segner^ ^ But the invention is doubtless Dr. Barkers. Their construction i shown in fig. 228, where C D is a vertical axis, moving on a pivot D, and carrying the uj)per millstone m, after passing through e opening in the fixed millstone C. Upon this axis is fixed a vertic^^ < tube T T communicating with a horizontal tube A B, at the extr-^B»t mities of which A, B, are two apertures in opposite directions. Wh— ^ water from the mill-course MN is introduced into the tube TT, H flows out of the apertures A, B, and by the re-action or count^^'- pressuro of the issuing water the arm AB, and consequently {9^e whole machine, is put iu motion. In order to understand how this motion is produced, let us svp* pose both the apertures shut, and the tube T T filled with water up to T. The apertures A and B, which ore shut up^ will be presseJ outwards by a force equal to the weight of a column of water whose height is T T, and whose area is the area of the apertures. Every part of the tube A B sustains a similar pressure ; but as these pres- sures arc balanced by equal and opposite pressures, the arm A B is at rest. By opcnhig the aperture at A, however, the pressure at that place is removed, and consequently the arm is carried round bv a pressure equal to that of a column TT, acting upon an area equal to that of the apei'ture A. The same thing happens on the arm TB; and these two pressures drive the arm A B round in the same direc- tion. This machine may evidently be applied to drive any kind of tnachinery, by fixing a wheel upon the vertical axis CD. 8. Mr. Runisey, an American^ and Mr. Segner, improved this machine, by conveying the water from the reservoir, not by a pipe, in greater part of which the spindle tuklis, but by a pipe which de- scends from a reservoir, as F, until it reaches lower than tlie arms A B, and then turns up by a curvilinear neck and collar, entering between the arms at the lower part, as shown in fig. 229. This greatly diminishes the friction. CHAP. VII.] AIB AS A MKCHANICAL AGENT. 297 9. Professor Playfair has correctly remarked that the raoving force becomes greater after the machine has hegan to move ; for the water in the horisoDta) arms acquires a centrifugal force, hy which its pres- sure against the sides is increased. When the machine works to the greatest advantage, the center of the perforations should move with the TelociQr — '\/hg^ where r is the radius of the horizontal arm, mearared from the axis of motion to the center of the perforation, and r J the radius of the perpendicular tube, g being put for the force of gravity, or 82 j^ feet. As 2 V r is the circumference described by the center of each per- foration, — =r is the Ume of a revolution in seconds. The quantity — y/hg is also the velocity of the effluent water; therefore, when the machine is worked to the greatest advantaee, the velodty with which water issues is equal to that with which it is carried borisontallv in an opposite direction ; so that, on coming out, it falls perpendicularly down. 10. The following dimensions have been successfully adopted; vis. radius of the arms from the center of the pivot to the center of the disdiarging holes, 46 inches; inside diameter of the arms, 3 inches ; diameter of the supplying pipe, 2 inches ; and height of the working head of water, 21 feet above the point of discharge. When the machine was not loaded, and had but one orifice open, it made 115 turns in a minute. This agrees to a velocity of 46 feet in a second for the orifice, being greater than the full velocity due to the head of water by between 9 and 10 feet ; the difference is due to the effect of the centrifugal force. The the<»y of this machine is yet imperfect \ but there can be no donbt of its utility in cases where the stream is small, with a con- stderable fidl. Mr. James Whiteland, a correspondent of the Franklin Joumaiy proposes to make the horizontal arms of the mill of a curved form, BQch that the water will run from the center to the extremity of the arms in a straight line when the machine is working. For the me- thod of constructing the curve, see Mechanics Magazine^ No. 499. It is very clear, however, that the additional efficiency of the machme will not be so great, by any means, as the inventor anti- cipates. Sbct» IL Air a$ a Mechanical Agent In the application of wind to mills, whatever varieties there may W in thrir internal structure, there are certain rules and maxims vrith 298 AIR A8 A MECHANICAL AGENT. [PABT II. regard to the position, shape, and magnitude of the sails, which will bring them into the best state for the action of the wind, and the pro- duction of useful effect. These haye been considered much at laige by Mr. Smeaton ; for this purpose he constructed a machine, of which a particular description is given in the Pkilosopkieal Tranm»' tions^ vol. 51. By means of a determinate weight it carried round an axis with an horizontal arm, upon which were four small movemble sails. Thus the sails met with a constant and equable blast of air ; and at they moved round, a string with a weight affixed to it was wound about their axis, and thus showed what kind of size or construction of sails answered the purpose best. With this machine a great number of experiments were made : the results of which were as fol- lows : — (1.) The sails set at the angle with the axis proposed as the best by M. Parent and others, viz. 55% was found to be the worst pro- portion of any that was tried. (2.) When the angle of the sails with the axis was incrcaiied from 72° to 75°, the power vths augmented in the proportion of 31 to 45; and this is the angle most commonly in use when the sails are planes. (3.) Were nothing more requisite than to cause the sails to acquire a certain degree of velocity by the wind, the position recommended by M. Parent would be the best. But if the sails are intended, with given dimensions, to produce the greatest effect possible in a given time, we must, if planes are made use of, confine onr angle within the limits of 72° and 75°. (4.) The variation of a degree or two, when the angle is near the best, is but of little consequence. (5.) When the wind falls upon concave sails it is an advantage to the power of the whole, though each part separately taken tihonld not be disposed of to the best advantage. (6.) From several experiments on a large scale, Mr. Smeaton has found the following angles to answer as well as any. The radius is supposed to be divided into six parts ; and ^th, reckoning from the center, is called 1, the extremity being denoted 6. No. Angle with th??Si2^ 1 72° IS'' 2 71 19 3 72 18 middle 4 74 16 5 771 121 6 83 7 extremity. (7.) Having thus obtained the best method o£ weat^erin^ the sails, t. e. the most advantageous manner in which they can be placed, our author's next care was to try what advantage could be deriTed from an increase of surface upon the same radius. The resolt obtained was, that a broader sail requires a larger angle ; and when the shI (Nil proper lur ciiiiirgtfu hmia ; ii uciiig luuiiu in prauucc umi uiis bould rather be too little than too much exposed to the direct of the wind. nc have imagined, that the larger the sail the greater would be iwer of the windmill, and have therefore proposed to fill up the area ; and by making each sail a sector of an ellipsis, acconling Parent's method, to intercept tlic whole cylinder of wind, in to produce the greatest effect possible. From our author's iments, however, it appeared, that when the surface of all the zeeedcd seven-eighths of the area, the effect was rather dimi- I than augmented. Hence he concludes, that when the wliolo er of wind is intercepted, it cannot then produce the greatest for want of proper interstices to escape. t is certainly desirable," says Mr. Smeaton, ^' that the sails of nills should be as short as possible ; but it is equally desirable, be quantity of cloth should be the least that may be, to avoid >e by sudden squalls of wind. The best structure, therefore, "ge mills, is that where the quantity of cloth is the greatest in a circle that can be: on this condition, that the effect holds out ^portion to the quantity of cloth ; for otherwise the effect can Rented in a given degree by a lesser increase of cloth upon a radiua than would be required if the cloth was increased upon me radius." ) The ratios between the velocities of windmill sails unloaded, rhen loaded to their maximum, turned out very different in dif- ; experiments; but the most common proportion was as 3 to 2. aeral it happened that where the power was greatest, whether enlargement of the surface of the sails, or an increased velocity ! wind, the second term of the ratio was diminished. ) The ratios between the least load that would stop the sails le maximum with which they would turn, were confined be- that of 10 to 8 and 10 to 9 ; being at a medium about 10 to Old 10 tn 0. or nhnnt A to /i; thnnrrh nn th<» whnio if orkrMMii*o«1 300 SUSATON's bulbs fob windmills. [pAHT 11. mils in ten or fifteen minutes ; and, from the length of the arms from tip to tip, has computed, that if an hoop of the same size was to ran upon plain ground with an equal velocity, it would go upwards of thirty miles in an hour. (11.) The load at the maximum is nearly, but somewhat less than, as the square of the velocity of the wind ; die shape and position of the sails being the same. (12.) The effects of the same sails at a maximum are nearly, but somewhat less than, as the cubes of the velocity of the wind* (13.) The load of the same sails at a maximum is nearly as the ^ squares, and the effect as the cubes of their number of turns in a^m given time. (14.) When sails are loaded so as to produce a maximum at m^ given velocity, and the velocity of the wind increases, the load con— <tf tinuing the same; then the increase of effect, when the increase oft-^s the velocity of the wind is small, will be nearly as the sqnares oft^-o these velocities: but when the velocity of the wind is double, th^^: effects will be nearly as 10 to 27|; and when the velocities com— jC3 pared are more than double of that where the given load produces i^ maximum, the effects increase nearly in a simple ratio of the velocit)^-' of the wind. Hence our author concludes, that windmills, snch wm^ the different species for draining water, &c, lose much of their effecft-^ by acting against one invariable opposition. (15.) In sails of a similar figure and position, the number of tium^ ^ in a given time will be reciprocally as the radius or length of the^ ^m sail. (IG.) The load at a maximum that sails of a similar figure ■w i j position will overcome, at a given distance from the center of motioo0^ will be as the cube of the radius. (17.) The effects of sails of similar position and figure are as tb« square of the radius. Hence augmenting the length of the sail with- out augmenting the quantity of cloth, does not increase the power; because what is gained by length of the lever is lost by the slownen of the motion. Hence also, if the sails are increased in length, the breadth remaining the same the effect will be as the radios. (18.) The velocity of the extremities of the Dutch sails, as well us of the enlarged soils, either unloaded or even when loaded to a maximum, is considerably greater than that of the wind itself. This appears plainly from the observations of Mr. Fciguson, already re- lated, concerning the velocity of soils. ■ (19.) From many observations of the comparative effects of aaih of various kinds, Mr. Smeaton concludes, that the enlarged sails are superior to those of the Dutch construction. (20.) He also makes several just remarks upon those windmills which are acted upon by the direct impulse of the wind against aaila fixed to a vertical shaft : his objections have, we beliere, been joa* tified in every instance by the inferior efficacy of these horiaontal mills. ^* The disadvantage of horisontal windmillsj" he remarks, ^ does CHAP. VII.] coulomb's EXPBRIMBNTS. 301 not consist in this, tliat eacb sail, when directly opposed to the wind, is capable of a less power than an oblique one of the same dimen- sions ; but that in an horizontal windmill little more than one sail can be acting at once : whereas in the common windmill, all the four ad together ; and therefore, supposing each Tane of an horizontol windmill to be of the same sise with that of a vertical one, it is mani- fest that the power of a vertical mill ^nll be foar times as great as that of an horizontal one, let the number of vanes be what they will. This disadvantage arises from the nature of the thing ; but if we oonsider the farther disadvantage that arises from the difficulty of getting the sails back again against the wind, &c., we need not won- der if this kind of mill is in reality found to have not above one- eighth or one-tenth of the power of the common sort ; as has ap- pMred in some attempts of this kind." coulomb's experiments. M. Conlomb, whose experiments have tended to the elucidation of many parts of practical mechanics, devoted some time to the subject of windmills. The results of his labours were published in the Memoirs of the Paris Academy for 1781. The mills to which he directed bis attention, were in the vicinity of Lille, and were, in fact, oil mills. From the outer extremity of one sail to the corresponding extremity of the opposite sail, was 70 feet, the breadth of each sau •J feet, of which the sail-cloth when extended occupies 5j feet, bemg attached on one side to a very light plank ; the line of junc- tion of the plank and of the sail-clod), forms, on the side struck by the wind, an angle sensibly concave at the commencement of the sail, but diminishes gradually all along so as to vanish at the remoter extremity. The angle with the axis, at seven feet from the shafl, is 60*, and it increases continually so as to amount to nearly 84"^ at the extremity. The shaft upon which the sails turn, is inclined to the horizon, in different angles in different mills, from H'' to 15^. Coulomb infers from his experiments, (I.) That the raUo between the space described by the wind in a second, and the number of turns of a sail in a minute, is nearly con- stant, whatever be the velocity of the wind ; the said ratio being about 10 to 6, or 5 to 3. (2.) That with a wind whose velocity is 21^ feet per second, the quantity of action produced by the impulsion of the wind is equiva- lent to a weight of 1080 pounds avoirdupois raised 270 feet in a minate; the useful effect being equivalent to a weight of 1080 pounds raised 232 feet in the same time : whence it results that the quantity of effect absorbed by the stroke of the stampers, the fric- tion, &c., is neariy a sixth part of the quantity of action. (3.) Suppose one of these mills to work eight hours in a day, Coolomb regards its daily useful effect as equivalent to that of 1 1 horses working at a walking- wheel, in a path of the usual radius. (4.) It is obaemble, that in most windmills the velocity at the 802 THE STEAM ENGINE. [pART II. extremity of the sails is greater than tbat of the ^iiid. In some coses, indeed, these velocities have been found in about the ratio of 5 to 2. Now, it is evident that the impulsion of a fluid a^inst any surface whatever, can only produce pressure, or mechanical effect, when the velocity of the surface exposed to the impulse is less than that of the fluid; and that the pressure will be nothing when the velocity of the surface is equal to, or greater than that of, the fluid. Indeed, in the latter case, the pressure may operate agaimt the motion of the sails, and be injurious. It is desirable, therefore, in order to derive from a windmill all the effect of which it is sus- ceptible, so to adjust the number of the turns that the velocity of the extremity of the sails bhall be IcMy or, at most, equal to that of the wind. It would be highly expedient to make comparative experiments on windmills, with a view to the determination of that velocity of the extremity of the sails which corresponds with the maximum of effect. If v denote the velocity of the wind in feet per second, t tbe time of one revolution of the sails, A the angle of inclination of tbe sails to the axis, and D the distance from the shaft or axis of rotation the point which is not at all acted on by the wind, or beyond whictx the sail-cloth ought to be folded up ; then theoretical considerations supply the following theorem, viz. : — * D = '1092 ft? tan A. Ex. Suppose 9 = 30 feet per second, t =2*25 seconds, an</ A = 75°; then D = -1092 X 30 X 2-25 x 373205 = 27-509 feet This result agrees nearly with one of Coulomb's experiments, in which the velocity of the wind was 30 feet per second, the fiails made 17 turns in a minute, and they were obliged to fold up more than 6 feet from the extremity of each sail, which were 34 feet long, to obtain a maximum of effect. The angle A at that distance from the tip of the sail was 75° or 76°. Sect. III. Mechanical Agents depending upon Heat: — the Steam Engine, In tbe steam engine the moving power is derived from tbe yapoiir produced from water, by the action of heat. Vapour is formed from water under ordinary circumstances at its usual temperature, wbat^ ever that may be; but the rapidity with which it is formed, and its elastic force, vary with the temperature and the pressure to which the water is subjected. In order to a proper understanding of this subject, it will be necessary to explain the exact meaning of a few of the terms em- iniiy. ine uensiiy or specinc gravuy or sieam, is, uiereiore, Iv as its specific volume. Tlie pressure^ tensiouy or ekutic f the steam, is the force expressed in pounds which it exerts every square inch of the interior surface of any vessel con- it. en water contained in an open vessel is exposed to any source , its temperature becomes elevated, and a portion of the water erted into vapour and passes otf in that form ; this elevation les until the temperature (as indicated by the thermometer) ! 212% when the vapour is formed with such rapidity as to e a considerable commotion in the water, which is then famili- id to boil. Notwithstanding, however, that fresh quantities of « (as before) being poured into the water, no further elevation emperature above 212^ can be produced so long as the vessel 8 open. But, if we now close the vessel, so as to prevent the from escaping as it is formed, the sensible temperature of the will again begin to rise, and will continue to do so as long as It is applied ; it will also be found that the temperature of the in the upper part of the vessel will always be the same as that water : and further, that as its temperature increases, so will *ic force, or the effort which it makes to escape, also increase ; g found that steam contained in a closed vessel in contact vater and exposed to any given temperature will, under all stances, be of the same density and have the same pressure, lowledge of the nature of heat, of the precise mode in which "mtes in the production of steam from water, of the species of lation between the heat and the particles of the water, and of uige (if any) which the particles of the water undergo in their sion into steam, is so imperfect, that we are not able, by any JdouB or reasoning, to arrive at the law which subsists between isible temperature and the pressure of steam in contact with ; as, however, the determination of the relation which they ■\ ^€uA\ tf\f)iAti ia Aoeonfial f/k orrivini* of nrsTrt^nt rAonUo in /\tii* 304 THB STEAM BNOINB. series, are the following, which have been collected bj Pambour. Mil III Author. Value of p = the preuuxe In lbs. per square inch. Value of f = the 1 bjrFahrenl Below 1 atmofphere From 1 to 4 From 4 to 50 Southern De Pambour {.SflSS.} (III.) / 98-806 -h t y \ 198MS / (V.) (•S69704 + '0060091 1)* (11.) 15$-72S6 (p - -OiMi (IV.) 198*062 ji* - (VI.) 14««1 p* - As these formula are of universal importance in all in nected with steam, whether as applied directly to the bU or to other purposes, such as warming buildings, &c., anc peculiar form having fractional or very high powers and comes necessary to employ logarithms in their applicatic express them logarithmically ; and, for the convenience o are not conversant with algebraical formulae, we shall reduce the logarithmic formulss to verbal rules, which we trate by a few examples. Expressed then, logarithmically, Southern's formulae be I. •.. Log(j» — -0494.8)=: 5-13 log(51-3 +0 — 11 II. ... Log (t -f 51-3) = 2-1923601 + ^^^, ,^ 5*lo De Pambonr*s — III. ... hogp = 6 log (98-806 + — 13-7873772; IV. ... LogC^ + 98-806) = 2-2978962 -f logp VI. ... Log (i + 39-644) = 2-1672906 + And those of Dulong and Arago — V. ... LogjD = 5 log (-269704 ± '0068031 0; logp Or reduced to verbal rules : — I. To find the pressure by Southerns formula. — Add temperature of the steam, multiply the logarithm of i 5*13, and from the product subtract 11-2468073, the ret be the logarithm of a number, to which, if we add -049 will represent the pressure in pounds per square inch. m.] TBS BTEAU EKQIVM. 305 npie. Required the pressure of steam at the temperature of Log of (150 -f 51-3 = 201-3) = 23038438 Moltipljring bj 5-13 11-818718694 SubtractiDg 11-246807300 Logof 3-73174 = -57191 1394 •73174 — -04948 = 3 78122 = the pressure required. fV Jind tke temperature hy SoutherriB formula,— From the \ in pouods subtract *04948, and divide the logarithm of the br bj 5-ldy to the quotient add 2*1923601, and the sum will ogarithm of a number, which, if we subtract 51*3 from it, fesent the temperature of the steam. 1^. What is the temperature of steam whose pressure is per square inch ? Log of (10 — -04948 = 9-95052) = 9978458 Dividing bj 5*13 •1945118 Adding 2-1923601 Log of 243-70 = 2-3868719 ••. 243-70 — 51*3 = 192*4, the required temperature. To/wrf the pressure by De Pamhour 8 formula, — Add 98 806 temperature, and 6 times the logarithm of their sum, minus 3772, will be the logarithm of the required pressure. ^pk. What is the pressure of steam at 247° ? Log of (247 + 98-806 = 345806) = 25388325 Multiplying by 6 15-2329950 Subtracting. 13*7873772 Log of 27901 (pressure required) = 1*4456178 T^iind the temperature by De Pambour'i formula.— Divide ■ithm of. the pressure by 6, and to the quotient add •t, the sum will be the logarithm of a number, which, if we •8*806 from it, will equal the temperature of the steam. 806 THB 8TBAM SNQINB. [PIBT Example. Required the temperature of steam having a pi of 35 Ihs. Log of 35 = 1*5440680 Dividing l>y... 6 •2573447 Adding 2-2978962 Log of 359 121 = 2 5552409 .-. 359-121 — 98-806 = 260215, the required temperature. V. To find the pressure hy Dulong and Arago*s formula.— H '^ dply the temperature hy -0068031, and to the product add -2697^ then 5 times the logarithm of their sum will he the logarithm of W pressure. Example. What is the pressure of steam having a temperature 330°? Log of (0068031 X 330 + -269704 = 2-514727) = •4004909i Multiply by S Log of 100-566 (pressure required) = 2^02454g VI. To find the temperature by DuiUmg and Aragos formula Divide the logarithm of the pressure by 5, and to the quotient 6 2-1672906, and the sum will be the logarithm of a number whicbi we subtract 39*644 from it, will equal the temperature. Example. What is the temperature of steam at the pressure 120 lbs.? Log of 120 == 20791812 Dividing by ... 5 •4158362 Adding 2-1672906 Log of 382-937 = 2-5831268 . - . 382-937 — 39*644 = 343-293 = the temperature iequire<3 In any boiler, or other evaporating vessel, if the source of ft be quite uniform, so that exactly equcJ portions of heat are giveim to the water in equal intervals of time, it will be found that the t of increase in the sensible temperature of the w%^r will dimihi^ as the sensible temperature itself increases; that is, that the Uaof rature of the water will rise (for example) from 60'' to 70'' in A heat thus required to convert any given weight of water into diminishes as the temperature of the water increases, and is that the sum of the sensible temperature and the latent heat is I constant and equivalent to 1 170° of Fahrenheit's thermometer, this circumstance two important consequences follow, viz., that 1 weight of steam, whatever may be its pressure and sensible rature, will really contain exactly the same amount of heat, ill therefore require precisely the same quantity of fuel for its •sion from water at 212° to steam at the given density, what- hat may be ; or in other words, that the same weight of fuel is ed to evaporate a given weight of water, and convert it into whether it be contained in an open vessel, or closed and sub- to any pressure, and this result has been amply confirmed by ment. The second consequence is, that if a quantity of steam larated from the water with which it was in contact, and then •d to expand into a larger space, or by pressure compressed into ler, 80 long as it does not lose any portion of its own heat, or e any fresh heat from surrounding bodies, its sensible tempera- ill always be precisely such as it would have been at the same re when in contact with the water from which it had been ced. And as in the steam engine, under ordinary circum- 8, and where the usual precautionary means are adopted for iting loss of heat by the steam in its passage to the cylinder, mperature may without any practical error be supposed to suf- loss from those causes ; it results from the property which we 'xplained above, that the temperature of the steam in the cylin- the engine and its pressure will always bear the same constant ►n to each other that we have already shown they do while in con- ith the water in the boiler ; and, therefore, the formulee and already given may be applied to determine either the tempera- r pressure of the steam (one being known) in any part of its »B through the engine. len water contained in a closed vessel is subjected to the con- I action of heat in the manner we have already supposed, the ireture of the steam, together with that of the water, gradually «e8y and fresh quantities of water are converted into steam; rom both these causes that the elastic force or pressure of the increases with the elevation of its sensible temperature. It "esults, that as fresh quantities of water are converted into , while the space which contains it is but very slightly increased, X 2 308 THE STEAM ENGINE. [PART II. the density of the steam must rapidly increase as the temperature rises, and its specific volume being inversely as its density, must become proportionally diminished. As the real quantity of water contained in the steam which passes through the engine, (and which depends upon the specific volume of the steam,) is one of the ne- cessary elements required in the calculation of the power of a steam engine, it becomes of importance to investigate the changes which steam undergoes in its density with any change in its temperature or elastic force. Let, therefore, V be the specific volume of steam at any tempera- ture t; then, since steam as well as all other aeriform fluids expand -^Q th part of their bulk at 32° for every degree that their sensible temperature is raised, putting V ^ for the specific volume of steam at 32°, we have V^ (f-32) 480 This formula supposes that no change has taken place in the pressure of the steam during its change in temperature, but as we have just shown that during all the changes which the steam under- goes in its passage through the engine, there is a constant relation between the sensible temperature of the steam and its pressure ; it therefore becomes necessary to determine in what way the specific volume of the steam depends upon its pressure. Now, when steam is separated from the water which produced it, and enclosed per se in any vessel, it then follows the same law relative to its density as any other aeriform fluid. This law has been already given at page 279, and is as follows, viz. : — that wliile the temperature re- mains the same, the elasticity of the steam, or its pressure against the sides of the vessel containing it, varies inversely as the space which it is made to occupy, or, directly as its density. Therefore, if P be put for the pressure of steam at 212% when its specific volume equals V, and p^ be put for its pressure, supposing its temperature unchanged but that its specific volume has become o, then, P : jt>j : : V : V P .-. t?=V — . Pi If, however, t^ represent the temperature which steam of the pressure p^ should have, and v^ represent its specific volume at that temperature, we shall have ^ * 480 ^ ^ 480 ' then, substituting the value of v obtained above, and reducing in respect of r^, we obtain _ P 448jM, '"''' p,' 660 • CHAP. VII.] TUB STEAM ENOINS. 309 Now, it is found that V, or the specific volume of Bteam at 212% is 1700, that is, steam at 212° occupies just 1700 times the space that the water from which it was formed occupied, and in this state its pressure, or P, is equal to one atmosphere, that is, thirty inches of mercury, or 14*706 lbs. on the square inch. Therefore, substituting these values of V and P in the preceding formula, we obtain 16969 4-37-879^, ^.,.,, Pi = ' (VII.) Pi By this formula it appears to he necessary that both the tempera- tore and pressure of the steam should be known, in order to arrive at its specific volume ; but we have already shown that the tempera- ture and pressure of the steam, both in the boiler and in its passage through the engine, preserve an invariable relation to each other, so that one can always be expressed in terms of the other; and we can therefore, by combining this formula with those already given, eliminate t^ altogether, and thereby obtain the value of e?. in terms of /?j only. But although the three sets of formulae whicn we have given, taken separately express this relation for a certain portion of the scale of pressures, with a sufficient degree of accuracy for all practical purposes, they will not admit of being correctly employed beyond the limits assigned to each. As, however, in the investiga- tion of the changes which take place in the pressure of the steam in the cylinder of the engine, it becomes essential to have some general formula which shall express with equal exactness this relation be- tween the specific volume and the pressure, for all temperatures and pressures wfiich can occur in the working of the engine, none of the foregoing formulae are sufficiently comprehensive to serve for this purpose; since, for example, the steam in an expansive engine may pass from a pressure of five or six atmospheres to that of one and a half to two atmospheres, which case would require the employment of both formulee (VI.) and (IV.), in order to discover the corresponding changes which had taken place in its specific volume. In order to remove this objection, Navier has proposed the follow- ing general formula, which expresses the specific volume of steam, in terms of its pressure only, and which, from the simplicity of its form, is peculiarly well adapted for the purposes of calculation 4^ ; it is v = ? (VIII.) in which v is the specific volume of the steam, having the pressure J} in lbs. per square foot; n and q are constants to be determined by * The following formula, derived from an expression for the relation between the pressure and temperature, given by the writer of the article on the Steam Engine, in the Encyclopaidia Metropolitana^ gives the specific volume of steam for all pressures above one atmosphere with singular exactness, but its complicated form prevents its being applied in the subsequent investigation. _ 12664-24 80861 6 ^ - p "^ p •"*"** * 310 THE STEAM ENGINE. [PABT IT. experiment, and the values of which, according to De Pamhour, should he as follows, viz. : — For condensing engines, n = -00004227, and q = -000000258 ; and for non-condensing engines, n = -0001421, and q = -00000023. The reason of the distinction being made between condensing and non-K:ondensing engines, is that the first values of n and q are found more correct for low temperatures, and the second values for high temperatures. If we represent by S the space actually occupied by a given weight of steam at the pressure /?, and whose relative volume is o, and by S j the space which the same weight of steam will occupy at the pres- sure /7|, and whose relative volume is v^, we have the following pro- portion : — V : v^ :: S : Sj ; and substituting for v and v. their values in terms of p and p^y bs derived from equation (VIII.) we have 1 1 n + q p n + qp^ ' and reducing in respect of/?, we obtain ^=:-'G +'.)-? "^•> Having thus explained the mechanical properties of steam, as far as is requisite for developing the theory of the steam engine, we shall conclude the subject by giving a table of the temperature and specific volume of steam for pressures varying from 5 lbs. on the square inch to 1 atmospheres ; the fourth column of which is cal- culated by FormulfiB (II.), (IV.) and (VI.), and the fifth colomn by Formula (VII.) CHAP. VII.] THB STEAM BNGINB. 311 TABLE OF THB PBB8SURB, TBMPBRATURB, AND SPECIFIC VOLUME OF FBOM 5 TO 150 POUNDS ON THB SQUARE INCH. STEAM, II II ll k h 1 M li II li h H =1 III iilii T20 5 161-4 4617 7056 49 281-0 664 Mi 6 169-2 3896 7200 50 282'3 563 1008 7 1760 3376 7344 51 283-6 643 1152 8 1820 2983 7488 52 284-8 684 1296 9 187'5 2673 7632 68 286-0 625 1440 10 192-5 2426 7776 64 287-2 615 1584 11 1974 2221 7020 65 288-4 607 1728 12 201-3 2050 8064 66 289-6 499 1872 13 205-9 1905 8208 67 200-7 490 2016 14 2001 1778 8462 68 291-9 483 1 2160 15 218-0 1669 8696 59 293-0 476 i 2S0i 16 216-4 1673 4 8640 60 293-9 468 2448 17 219^6 1487 8784 61 294-8 461 25^3 18 222-6 1411 8928 62 296-9 454 2736 19 2255 134S 9072 63 297-0 448 2880 20 228-3 1281 9216 64 298-1 441 aos4 21 2310 1225 9360 65 299-1 435 8168 22 233-6 1173 9504 m 300-1 1 429 3312 23 2360 1127 9643 67 301 '2 424 34 Se 24 238-4 1083 9792 ^8 302-2 418 seoo 25 ; 240-7 1043 9930 69 SOS-iJ 412 3744 28 243*0 1010 10080 70 - 304-2 407 3888 27 2461 972 10224 71 306-1 402 4m2 28 247-2 940 10363 72 306'1 897 417e 29 249-2 911 10612 78 307-1 392 2 4320 30 251-2 882 10656 74 308-0 386 4484 31 253-1 857 5 108(90 75 308-9 382 1 4608 32 255-0 832 10944 76 309-9 878 4752 38 256-8 809 11088 77 310-8 373 4898 34 258-6 787 11232 78 311-7 369 5040 35 260-3 767 11376 79 312-6 366 £184 36 262-0 747 11520 80 313-6 361 dms 87 263-7 729 11664 81 314-4 357 6472 38 2653 711 11808 82 315-2 363 5616 39 266-9 694 11952 83 816-1 349 5760 40 268-4 678 12096 84 816-9 345 5904 41 269-9 663 12240 86 317-8 841 6048 42 271-4 649 12384 86 318-6 338 6192 43 272-0 636 12628 87 319-4 334 6336 44 2743 622 12672 88 320-3 331 3 6480 46 276-7 609 12816 89 321-1 327 6624 46 277-1 697 6 12980 90 321-9 324 6768 47 278-4 635 13104 91 3227 321 6912 48 279-7 574 18243 92 3236 318 812 THE 8TSAM ENGINE. [part 11, 1 II 11 11 111 iHill 11 1 II e II no ^1 hi 271 13392 93 3:24*3 315 1 15S40 3307 13536 H 325 312 1 16560 lis 340-1 2m 130J>0 95 325-8 3oa @ 17'iaO 120 343'S 250 13924 96 32fltf 30(J ISOOO 125 346-4 241 1396S ^7 327-3 303 18720 130 3496 233 HI12 38 328-1 mo » 10440 135 35:2-4 225 lit25fi m 328-8 297 201S0 140 355-3 217 144 im urn 32P-6 2i^S 20830 145 35S1 211 7 16120 1(15 333-2 28-2 10 1 21600 150 360*8 204 GBNEBAL DESCRIPTION OF THE MODE OP ACTION OP THE STEAM ENGINE. Before proceeding to develope the general theory of the steam engine, or to investigate the mode of action of the steam in the cylinder, it will be well to enter into a general description of its construction and mode of action. In this preliminary description we shall only notice two forms of engine, namely, the non -condensing expansive engine, and the condensing engine without expansion. Plate VII. exhibits a sectional elevation of a condensing steam- engine of eight horse power, with its boiler attached. The steam being generated in the boiler A, is conveyed by the steam-pipe C into the valve- box or chamber I, which is shown upon a larger scale in fig. 1, plate IX.; from this chamber there are three passages, one (a) communicating with the top of the cylinder, the other (b) with the bottom of the cylinder, and the third (c) with a yeasel to be afterwards described, called the condenser. These passages are covered by a sliding valve, </, so formed that when in the position shown in fig. 1, all three of the passages are closed, but if it be moved downwards, and brought into the position shown in fig. 2, the upper passage a is made to communicate with the steam cham- ber I, while b and c are made to communicate with each other ; but if it be moved upwards, and brought into the position shown in fig. 3, then the lower passage c communicates with I, and a and b with each other. The action of this valve being understood, if we snppose it to be in the position shown in fig. 2, we must immediately per- ceive that the steatn, which has been conveyed into the chamber I by the pipe C, will pass by means of the passage a into the top of the steam cylinder D, and by its pressure on the piston E give motion to it, and cause it to descend. This piston is connected by a rod e with a beam capable of turning about the center F, and the other end of which is connected by the rod ^ with a crank hy upon the shaft of which is fixed the fly-wheel O. It is therefore evident, ihftt any CHAP. VII.] THE STEAM ENGINE. 313 motion of the piston £ will by means of the beam and connecting rods be immediately communiatcd to the crank shaft h, and cause it to revolve. Upon this shaft a contrivance termed an excentric (H) is fixed ; this is nothing more than a circular disc, which has the shaft or axis passing on one side of its center, the effect of which arrange- ment is to cause any point in its circumference to move nearer to or further from its center as it is made to revolve, by which means a reciprocating motion is given to the connecting rod k, which commu- nicates through the beam /, and rods m and n, to the slide valve d already described. Now, when the piston E, by the continued pres- sure of the steam upon its upper surface, has been forced to the bot- tom of the cylinder D, this valve d will, by means of the excentric, have been moved upwards, and brought into the position shown in figure 1, in which the communication between the boiler and the cylinder is closed, so that no more steam can be admitted to press upon the piston £. The downward motion, however, of the piston liaving been communicated, in the manner already described, to the fly.wheel O, the momentum which it has thereby acquired causes it to continue its motion, and as it carries round the crank ^, and the excentric H will produce the following twofold effect ; first, it will, through the instrumentality of the beam F, and connecting rods, e and^, reverse the motion of the piston, and cause it to commence its ascent in the cylinder; secondly, it will, by means of the excentric and system of rods attached to it, cause the valve d to slide upwards, to- wards the position shown in fig. 3. As soon as this motion of the slide valve commences, t)ie communication between the passage c and the box I being opened, the steam will now pass to the bottom of the cylinder, and there pressing on the lower side of the piston, will cause its reascent. We must now, however, notice what becomes of the steam which already occupies the upper portion of the cylinder. By reference to fig. 3, (which it must be recollected represents the position in which the slide valve is now supposed to stand,) the upper part of the cylinder will be seen to be in immediate communication, by means of the passages a and b^ and the pipe O, with the condenser M, which is a cylindrical vessel entirely surrounded with cold watet, and which also has a small jet of cold water constantly playing into iL If we now suppose the whole of the air to have been previously expelled from the vessel M, (the manner of doing which will be pre- sently described,) so that nearly a perfect vacuum is formed within it, it will readily be understood that the moment the communication is opened between it and the upper part of the cylinder, the steam con- tained in the latter will rush into this vacuum with a very great velo- city, and being there immediately condensed Jby the cold water playing into and also surrounding the vessel M, the vacuum will be preserved, and thus no resistance will be offered to the motion of the piston by the pressure of the steam upon its upper surface. As, however, the injected water, together with that resulting from the condeDBed steam, would in time fill the vessel M, an air pump, L, is 314 THE 8TBAM BNOINB. [PABT II. made to communicate with it, which being worked by a rod from the beam of the engine, always keeps the condenser empty. And fur- ther, as the heat lost by the steam would in a short time so far elevate the temperature of the water surrounding the condenser as to render it incapable of continuing properly to condense it, it be- comes necessary continually to change the water, with which object a pump, P, worked by a rod from the engine beam, raises cold water from a well or other source^ and pours a continued stream into the vessel surrounding the condenser, while an equal quantity of the warmer water is allowed to run off. Before the engine is set to work, the cylinder D, the condenser M, and the passages between them, are filled with common air, which it is necessary to extract To effect this, by opening the valves a communication is made between the steam-pipe C, the space below the piston in the cylinder D, the eduction-pipe O, and the condenser M. The steam will not at first enter the cylinder D, or will only enter it a little way, because it is resisted by the air ; but the air in the eduction-pipe O, and the condenser M, is forcibly driven before it, and this part of the air makes its exit through the valve N, called the suift valve, and which is kept covered with water. The steam-admission valve is now closed, and the steam already admitted is converted into water, in the manner already described, by the coldness of the condenser M, and by the jet of cold water which enters it from the well SS, in which the condenser is immersed. When this steam is condensed, all the space it occupied would be a vacuum, did not the air in the cylinder D expand, and fill all the space that the original quantity of it filled ; but by the repetition of the means for extracting a part of the air, the remainder is blown out, and the cylinder becomes filled with steam alone. In order that the connecting rod e may work freely, and yet possess the desirable property of being steam-tight, it passes through what is called a stuffing or packing box. This stuffing consists of some material which the steam will rather adapt to its office than injure; leather, which answers well for the stuffing or collars of machines never to be subjected to heat, will not answer here; hempen yarn is the material usually employed. The rod of the piston / passes through a stuffing box of the same kind as that of the piston £ ; and the pistons themselves are surrounded with stuffing. The cylinder D is surrounded by a case, to keep it from being cooled by contact with the external air. The extremity, or any given point removed from the center of the great beam, can describe only the arc of a circle; but it is necessary that the piston rod e should rise and fall vertically. An apparatus is therefore used, called the parallel joint, which is easily understood by inspection. By this means the rod e not only rises and falls perpendicularly, bat is perfectly rigid, and communicates all its motion to the great beam in each direction of its motion. The connectiivj rod g does not require the same contrivance, because it does not rise and fall per- pendicularly; its lower end, with the outer end of the crank, describ- CHAP. VII.] THB STEAM ENGINE. 315 ing a circle : it has therefore only a simple joint, admitting of this deviation. In order to communicate a rotatory motion to the fly-wheel, in- stead of the crank may he used a contrivance giving twice the rapidity to the fly. For this purpose, on the outside of the axis of the fly, irhere the crank is shown in the plate, a small toothed wheel is fixed, and can only he moved with the fly : at the extremity of the rod g^ and on that side of it which is next the fly-wheel, another toothed wheel is fixed, in such a manner that it cannot turn round on its axis, hut must rise and fall with the rod to which it is attached. These two wheels work in each other, and that attached to the connect- ing rod cannot leave its fellow, because their centers are connected by a strap or bar of iron. When, therefore, the connecting rod rises, the wheel upon it moves round the circumference of the wheel upon the axis of the fly. By this means the fly makes an entire revolution for every stroke of the piston, and some mechanics are apt to think that they are great gainers by such an arrangement : the contrivance is certainly el^ant, but with respect to utility, the fact is, that a crank is preferable ; for it is more simple, cheaper, and less likely to be out of order, while, if the fly be large enough to receive, with less velo- city, all the momentum that can be communicated to it, the efiect will certainly not be inferior. We now pass on to describe the non-condensing expansive engine shown in plate VIII., in which fig. 1 is a side elevation ; fig. 2 an end eleTation; fig.daplan; and fig. 4 plate IX., asection of the cylinder, show- ing the steam passages and valves. By a glance at this plate, the means by which the motion of the piston is transmitted to the crank shaft "mil be seen to be similar to that just described ; the real diflerence between them consists in there being no condenser and air pump in the engine which we are now describing, and in its having a second ezcentric by which the admission of steam to the chamber I from the boiler can be cut off at any desired portion of the stroke. The mode of admitting the steam to the top and bottom of the cylinder, alter- nately, by means of the slide valve d^ is precisely similar to that already described; but after the steam has done its duty in the cylinder, instead of passing into the condenser it escapes by the passage d, directly into the atmosphere. We have yet, however, to explain the use of the second cxcentric, and to show in what manner the steam is caused to expand in the cylinder, by which a saving is eflTected in the quantity of steam used by the engine. In the engine which we have just described, the steam is supposed to remain on during nearly the whole stroke of the engine, and to have therefore nearly the same pressure throughout the stroke ; but in that which we are now describing, the communication between the boiler and the cylinder can be closed at any period of the stroke that is desired, so that after the steam has been cut off, it then expands in the cylinder as the piston moves before it, and therefore presses upon the piston at each instant with a pressure due to its specific volume at that instant. The method of adjusting the exccntric rod, so as to cut off the 316 THE STEAM BN6INE. [pABT II. Steam from the cylinder at any desired period of the stroke, is shown in fig. 5, plate IX., and is as follows: — ah is a bell-crank lever, by means of which the motion of the excentric is communicated to the valve, the arc cd being the distance (termed its effective stroke) through which the valve has to be moved to cut off the steam entirely from the cylinder; then, if we make the angle gkk equal to the angle ckd^ and taking the diameter of the excentric's path, or double the distance between the center of the excentric and the center of the shaft in the compasses, apply them along the two dotted lines, the point marked 6 where they fall upon both, or where the angular distance is equal to the diameter of the excentric's path, will be the f)oint at which the excentric rod must be fixed to the arm h of the ever, in order that the steam may remain on during the entire stroke. Then, fig. 6, representing the excentric's path, if it be desired to cut off the steam at any portion of the stroke, as at ^, i, or J, we have only to measure such a proportion of the semi- circumference of the excentric's path, and taking the length of its chord in the compasses, apply them in the manner already described, to the angle gkh^ and the distances from k^ at which they severally fall upon both lines, will be the distances at which the excentric rod should be fixed when it is desired to cut the steam off from the cylinder at any such portions of the stroke. Or, if we put / for the length of the arm a of the lever, f for the effective stroke of the valve or the distance erf, ^ for the diameter of the excentric's path, d for the distance from k to the point at which the excentric rod should be attached to the arm h of the lever, and n the fraction of the stroke at which it is desired to cut the steam off from the cylinder, then /^8in(90«) E For example, if / equal 6 inches, t equal 1*5 inch, and 1 equal 3 inches, and we give to n the successive values *25, '5, "75, or suppose the steam to be cut off at J , \ and \ of the stroke, we shall have for the corresponding values of rf, , , C X 3 X -383 ^ ^ . , at J stroke d = z = **^ inches, 1*5 at ^ stroke d = = 8*5 „ 1*5 1 . , , ^ 6 X 3 X -924 and at } stroke d = = 1 1-1 „ 1*5 THKORV OF THE STEAM ENGINE. Before entering uj)on the general theory of the steam engine, it is but right to mention tliat the manner in which we purpose consider- ing the subject is tjimilnr to that first laid down by the Comte De Pambour, in his very able work * upon this subject ; and at the * The Theory of the Steam Engine, by Comte De Pambour. PubUalaed bj John M'eale, 1839. CHAP. VII.] THB 8TBAM ENGINE. 317 same time to acknowledge the great services which he has rendered b? his investigations, which have led to the adoption of a theory alike applicable to every form of engine, and working under all cir- cumstances. The propositions upon which his theory is founded, are thus stated by him at page 25 of his work :^— ^^ From what has been stated, it plainly appears that we ground all our theory on these two incontestable facts: 1st, that the engine having attained uniform motion, there is necessarily equilibrium between the power and the resistance; that is, between the pressure of the steam in the cylinder^ and the resistance against the piston, which furnishes the first relation *, Pj = R. And 2dly, that there is also a necessary equality between the production of the steam and its expenditure, which furnishes the second relation t, _ w8 P And these two equations suffice for the solution of all the problems." The manner in which Tredgold, and other writers upon this sub- ject, had determined the power of an engine, was by measuring the area of its piston, and then, assuming that the engine would move at a given velocity, and that the pressure in the cylinder would be identical with that in the boiler, they thought that the continual product of these quantities would give them the load which the engine ought to raise in a given time, and to a given height, pro- Tiding no loss had arisen from the friction of the machine, and other canses; and further, supposing that the portion thus lost always bore the same constant ratio to the whole, in the same cla.ss of engines, they determined arbitrary coefficients, by which they multi- plied the above product, and they conceived that the quantity thus obtained represented the actual effective power of the engine with sufficient accuracy for all praqtical purposes, although no notice ivhaterer was taken in the calculation of the evaporative power of the boiler. A little consideration will, however, show the inaccuracy of this method of calculation. The engine itself is not the moving power by which we produce the desired effects; the boiler is the part in ^vhich the power resides, and the engine is only the instrument or machine through which this power is transmitted, and by which it can be applied precisely at the point, in the mode, and with the * In whidi p, represents the pressure of the steam against the piston, per unit of surfaoe, and a represents the resistance of the load against the piston, divided in like manner per unit of surface. •f In which v is the velocity oi the piston under the resistance a, m the specific Tohime of the steam in the hoiler under the pressure p, or the ratio of Its Tohime to that of the water which produced ii^ s the volume of water er apo rafd by the boiler in a unit of time, and a the area of the cylinder. 318 THE STEAM BNOINB. [paR*^ velocity which we desire. A steam engine may, in this respect^ compared with a crane, by means of which a number of met^ enabled to raise a considerable weight ; bat here no power resid.ef the crane, it is merely the instrument through which the mcT^ i enabled to apply simultaneously their united energies, and to pro^i certain effects; but those men could produce an equal amon^int effect without the crane, although not with so much ease or con ^ei ence. And, in like manner, the power which can be transmitt^ the engine can only equal that which resides in the boiler. For, in the crane, the power exerted through it must depend on tl number of men by whom it is moved, so also in the steam engfiK must the power which it can exert, or the amount of work which i can perform, depend only upon the power of the boiler. iVot however, that we mean to assert, that any amount of power which it is possible to derive from a boiler could be transmitted and app/fecf by the engine, for, as the crane has been designed and proportfooed only for the production of a certain effect, and the application of a greater power than that required for its production might strain and injure the machine, so it is with the steam engine, every engine is constructed only to transmit safely a certain amount of power frotn the boiler to the working point, and the attempt to produce a grease effect might be attended by the derangement of the machinery. If a equal the area of the piston in square feet, and p^ ^ pressure of the steam in the cylinder, per square foot of surface, tl^ product, or a/?, will obviously represent the resistance which ^ engine is capable of overcoming; and if this be multiplied by ^ length of the piston's stroke = /j , we shall have the whole amo^ of work which the engine will perform in each stroke, or represent^ this amount by k'i , we shall have », = a/,/?, (X.) This equation, however, supposes the steam to act with C pressure pi during the whole stroke of the engine, and therefore on applies to engines in which the steam is not cut off until the end • the stroke, and which therefore do not work expansively. In order then to obtain a general expression which will indac the expansive engine also, let us suppose that /j represents only th portion of the stroke during which the steam is not cut off, in whi< case the above formula will still represent the work done by tl engine during such portion of the stroke. Now, if x equals tl length of a portion of the cylinder, equal in content to the stei passages and the space left at each end of the cylinder for the clei ance of the piston, we shall have a (/, + >^) for die actual space occ pied by the steam in the cylinder (having a pressure of />, ) at t moment when the steam is cut off; then, if/ represent the length of t stroke from its commencement, at any moment after the expansi of the steam has commenced, and p the pressure of the steam d to its altered volume, we shall have from formula (IX.) /, + X /w \ n CHAP. VII.] THB STEAM ENGINE. 319 Then, since the work done by the steam in the cylinder is equal to the pressare upon the piston multiplied by its area and the length of the stroke, or the distance that the piston has moved under that pressare, if A / represent the distance moved over by the piston after the steam has been cut ofif^ we shall have for the work performed daring such motion, a/7 a/, in which, substituting for p its value derived above, we have therefore, the whole amount of work done by the steam during its expansion in the cylinder will be represented by or putting w, for this amount of work, and /, for the total length of the stroke from its commencement, «nd int^rating between the limits of /, and /,, we have ». = a (/. + X) hyper. Wj^ (^ + ^ ») " f « (^' - ^•) - (^I) Now, if we add together the work done before expansion, (= t0„) and the work done after expansion, ( = «;„) and represent the whole work performed by the engine each stroke by W, we shall have W = w, + w,; in which, substituting for w^ and w, their values derived from equa- tions (X.) and (XI.), and reducing, we obtain W=a Q + ;».) [K + (/. + A)hyper. log \±^^ - ^ a /.....(XI/) Then, if P represent the pressure or resistance which the work W exerts against every square foot of the surface of the piston, W = Pa^, and therefore P/. = d + />.) {/. + (/. + X) hyper. logjf^} -]ir - (XII.) wbich ezpreanon becomes P=J», (XIII.) when l^=zli^ or when there is no expansion. These forrouls, then, enable us to determine the effects which would be produced by any engine working under given circumstances; hat it b^mes a matter of considerable practical importance to 320 THE STEAM BN6INB. [PART II. determine those values of the several quantities which shall produce the greatest mechanical effect with the least expenditure of steam ; or, in other words, the quantity of water evaporated by the boiler remaining constant, shall give to P its maximum value. In the high- pressure engine, working without expansion, this takes place when the pressure in the cylinder is as nearly as possible equal to that in the boiler. In the expansive engine, the same law holds with regard to the pressure at which the steam should be admitted to the cylinder to produce the greatest mechanical effect with any given expansion. But there is in every engine working by expansion a certain proportion of the stroke through which the expansion will produce a greater effect than through any other proportion, and this will be easily found after we have obtained expressions for Q, or the number of cubic feet of water which the boiler is capable of evapo- rating per minute. If V. equals the specific volume of the steam at the pressure /?, , at which it is admitted to the cylinder, we shall have for the num- ber of cubic feet of steam of that pressure, generated per minute, Q f 1 ; and if the engine makes » strokes per minute, each equal /j feet in length, »a(/j + ^) will equal the number of cubic feet of steam at the pressure p^ used by the engine per minute, and therefore from the second principle laid doivn by De Pambour*, we have Qr, = ar(/, H- >); then, if V equal the velocity of the piston in feet per minute, V we have V = i-/^ , or » = - ; and also from equation (VIII.) we have V, = ; therefore, substitutine these values above, we obtain Q aV(/, +x) n + qp^ I, whence we obtain for the value ofp j , _ l,Q n (XIV.) ^* ayV(/, +A) g We have also seen by equation (XIII.) that in an unexpansive engine /^j =: p, p being the resistance against each square foot of tlie piston's surface, arising from its load, from the friction of the engine, and from the pressure of the atmosphere, or imperfectly condensed steam on the other side of the piston. Now, taking the most simple case, or that of the stationary engine, if we put ^ for the resistance occasioned by the load upon each square foot of the piston's surface, J* ^^'^ ^^^ resistance arising from the friction of the engine when unloaded, and ^ ^ for the additional friction produced * See page 317. CHAP. YII.] THB STB AM BNOINB. 321 by the load ^ upon eyery square foot of the piston s surface, and also r to represent die pressure of the atmosphere or uncondensed steam upon each square foot, we have and substituting this value of jp^ in equation (XIV.) above, ";*»>+/^'=.-?vTi7T^-? <^^> wbich equation expresses the relation between all the several quan- tities for unesfpansive eneines. In exfMUuive engines 7^ represents only that portion of the stroke which IS performed before the steam is cut off, the whole stroke being equal to /^ ; therefore in this case ' = j- > cuid equation (XIV.) becomes '.=j?4rnr)-F <''^'' Now, from equation (XII.) we have for expansive engines, /, +(/,+x) hyper, log ^-^ ^^ •nd substituting for P its value determined above, reducing, and solving in respect of ^, we have Q(rV, + typer.log^j) /+. + ^ ( = ^1 ^ ^i, I M+^-^ - _ 1 ... (XVII.) Then, aboe the actual useful mechanical effect produced by the engine per minute equals ^ a V, we have for the power of the engine ^ ^(rTi + ''yp-'^°grT-D «v(/-H^4-f) «"^ — TUTf^ — 1 + * Now, in order to arrive at that value of /^ which will give the maadmom value to ^ aV, or, in other words, to ascertain at what |iortion of the stroke the steam should be cut off in order to produce the greatest effect, let us substitute for a V its value derived from eqvfttion (XIV.), whence by reduction we obtain 9(1 +f)V/, + X^ '*^ */, +x l^+?, n+qp^ J 322 THE STEAM ENGINE. [PART II. then, differentiating in respect of /^ and remembering that since the above expression is a maximum its differential will = 0, we have whence, reducing and solving in respect of j , we obtain (XVIII.) equal that portion of the stroke (the whole stroke being unity) at which the steam should be cut off to produce the greatest effect, with the least expenditure of steam. DESCRIPTION OP THE VARIOUS KINDS OP ENGINES, AND THE FORMULAS POR CALCULATING THEIR POWER. Although almost every manufacturer of steam engines adopts his own peculiar form of construction, by which a great diversity in their external appearance is occasioned, the principles upon which they act are similar in all engines of the same class; and, therefore, in attempting a classification of the different kinds of engines, we shall pay no attention to the details of their construction, but only regard the mode of employing the steam in the cylinders for the purpose of producing any particular species of motion. Proceeding, then, upon these principles, the following table represents at one view the various forms under which steam is at present employed as a moving power. Engines working | Non-condensing [ f^^'^, «m^/ expansion | Condensing. ^ Engines working with expansion r"-"- [c^^ j-^-«- ^ J Steam pressing only on the c;,i»io i.^:»» J t upper surface of the piston. V Single acting j | st^^„,^i„^ ^^U on the I Steam pressing only on the lower surface of the piston. Stationary non-eandenBtng EngineB without expansion. The mode of action of the steam in the cylinders of this class <- ^ " engines is precisely similar to that already described at page 31^^ and a view of which is given in Plate VIII., with the excepUon tbi^^ the steam is never cut off from the valve-box I, and therefore pr ess e d with an equal pressure upon the piston daring nearly the who^ff extent of the stroke. With regard to the mechanical coDatmction a^ this kind of engine, it depends in a great measure upon the fancy c^- the maker, or upon the purposes to which the engine is to be applied ^ there is perhaps a greater diversity in the form of this class (^'« engine than of any other. For particular examples of the method CHAP. VII.] THE STEAM ENGINE. 323 of coDStnictiDg all kinds of CDgines, we must refer to the last edition of Tredgold's elaborate work upon the steam engine, which, with its appendices, forms one of the most magnificent works ever published. The formulse for calculating the power of engines of this class may be immediately derived from equation (XV.) already given, by solving in respect of any of the quantities which are wanted. Thus, we obtain for the values of V^ Q, and ^, v= ^^. or, multiplying ^ by Vo, which then gives the actual work performed by the engine in each minute, we have We may reduce these formulae to a more practical form by insert- ing the numerical values of those quantities which are either constant, or sufficiently so for all practical purposes. Thus, t being in this case the pressure in pounds of the atmosphere upon each square foot of the piston's surface, equals 2118 lbs. ; for n and ^, we have the values already given for non-condensing engines at page 310, and assuming that x = *05 /„, ^ = '14, and y = 144, as adopted by De Pambour, the above formulae become ^ "■ a(6'95536 + 0027531 f)' Vg (6-95536 -f '0027531 e) ^ ■" 10000 ^aV = 3632268 Q — 2526-37 a V. Locomotive Engine working without expansion. The mode of action of the steam in the cylinder of a locomotive engine, does not in any way differ from that in the engine already described; the only essential differences of which we have to take notice being in the details of its construction, and in the nature of the work which it has to perform. Plate X. represents the longitudinal, and Plate XI. two transverse sections of one of the engines employed on the Great Western Rail- way. The principal peculiarity in the construction of a locomotive consists in the form of the boiler, which, in order to lessen the weight fo be moTed, is much smaller than the ordinary boiler for other Y 2 324 THB 8TBAM BNOINB. [PABT II. engines of the same power, and wbicb, therefore, requires a very intense heat in the furnace, and a large evaporating surface: the first of these is effected by emp]o3ring coke instead of coal, and pro- ducing rapid combustion by a strong draught; and the second bv causing the water entirely to surround the fire-box, or furnace, and by carrying the smoke and heated air from the same through a number of small tubes which are likewise surrounded witli water. Thus, it will be seen by an inspection of Plate X., and also of fig. 2, Plate XL, (which latter is a transverse section through tlie fire-box,) that the water not only covers its upper surface, but that it surrounds it on every side, leaving only a small space on one side for the opening, I, by which the stoker regulates the fire. The tubes aa^ to which we have already alluded, for convejang the smoke from the fire to the chimney, are ninety-one in number, and expose a very large heating surface to the water ; and in addition there is a hollow bridge, £, also filled with water passing through the center of the fire. The bars, X X, of the furnace, are so arranged, that when it is desired to put out the fire they can all be simultaneously lowered, allowing the burning contents of the furnace to fall upon the ground, or into a vessel placed to receive them underneath the engine. The sides of the fire-box, being flat, would be liable to be forced out by the pressure of the water and steam, to obviate which the sides are held together by short rivets, and the top is strengthened by a number of short girders ^, bolted to its upper side. The flat ends of the boiler are prevented from being forced out by the fire tabes aay which are secured at each end, and by tie rods which run from one end to the other; the other parts of the boiler derive snflicient strength from its cylindrical form. There are two safety-valves, one d, under the immediate inspection of the engine-driver, and which by means of a spring manometer Q, always indicates to him the pressure existing in the boiler ; the other L, is placed (as an additional pre- caution) beyond his reach, so that it cannot be meddled with, or have its pressure altered. A glass tube W, fig. 1, Plate XL, made to communicate with the boiler at both its upper and lower extremities, so that the water in it always stands at the same level as that in the boiler, serves to indicate to the engine-driver when the quantity of water falls short. An opening R, called a man-hole, is left in the upper part of the boiler, being covered with a screwed lid, in order to enable the interior of the boiler to be examined ; c is a whistle, so contrived, that the steam being allowed to rush forcibly through it— - produces a very shrill whistle, which is employed as a very generaL^ and useful means of signaling on railways. It will be obsenred that^ the boiler is surrounded by an exterior case, a small space being lefk:^ between and filled with air, which prevents in a considerable degress the loss of heat from radiation, that would otherwise take place. In consequence of the steam space in a locomotive boiler beings very limited, the steam is formed almost as it is used, and immedi-' atefy the communication with the cylinders is opened and the steaitt allowed to flow into them, a violent commotion in the water in thff CHAP. VII.] THE STEAlf ENGINE. 325 .boiler is occasioned by tbe sudden production of steam to supply its place, the result of which is, that a large quantity of water, in a state of minute mechanical division, is carried over with the steam into the cvlinders, and there gradually accumulating, would at length materi- ally interfere with the working of the engine. This effect, which is called primingy is to a certain extent obviated by a dome and steam- chest (H) heing formed at the top of the boiler, and by the entrance of the steam pipe G by which the steam is conveyed to the cylinders being carried to the upper part of this chest, and having its mouth turned upwards, by which the space for steam is somewhat increased, and more time is given for the separation of the water from the steam. The admission of steam to the cylinders is regulated by the Talve F, which the engine-driver can open or close by means of the handle d. The steam pipe has an expansive joint Z, which allows of a slight elongation or contraction of the pipe, in consequence of its varvms temperature. The steam pipe passes straight through the boiler mto the chamber in the fore part of the locomotive, termed the smoke-box, it there divides as shown in fig. 1, Plate XL, into two hrancbesX/y ^ch leading to one of the cylinders; the manner in which the admission of the steam alternately to the top and bottom of the cylinder is effected by the slide valve T, is precisely similar to that alr^y described at page 312. But in a locomotive it is neces- sary that Uie engine-driver should possess the means of making the engine revolve in either direction, in order to effect which there are two excentrics to each slide valve, fixed upon the crank shaft, oppo- site to each other, and so arranged that, by means of a lever within reach of the engine-driver, he can connect either excentric with the slide valve, and by reversing its motion also reverse that of the engine itself: only one of these excentrics (C) is seen in the plate. The motion of the piston n, is transmitted to the crank »«, by the piston rod o, and connecting rod r; the piston rod being made to move straight by the cross head B, moving in guides. After the steam has performed its office in the cylinders it is expelled through the pipe M, into the lower part of the chimney, where by its partial condensation it produces a powerful current of air through the fire and tubes aa. The water formed in the cylinder by condensation, or that which is brought over by priming, is allowed to escape by the cock P. The supply of water to the boiler is maintained by a small force-pump D, the plunger of which is attached to the cross head B of the piston rod, and which draws tbe water from the tender through the pipe O, and forces it into the boiler. The cylinders, bein^ entirely within the smoke-box, are always maintained at a considerable temperature, by which means very little loss results from condensation of steam in the steam passages or cylinders. The nature of the resistances to which a locomotive engine is subjected are somewhat different to those of a stationary engine, arising from the two following causes, viz., the resistance occa- sioned against the piston, by the ejected steam being blo\^ii forcibly throDgb tbe contracted pipe M into the chimney, for the purpose of 326 THB STEAM ENGINE. [PART II. producing a draught in the fire, as already described ; and the resistance which the air occasions to the progressive motion of the train. The former of these, the Comte de Pambour, by his experiments, has ascertained to increase directly as the velocity of the piston, and to amount upon an average to about 1*75 lbs. per square inch of the piston's surface, when moving with a velocity of 150 feet per minute, or putting tt ^ V for this pressure per square foot, at the velocity V, we have ^1 : 1-75 X 144 : : V : 150; .-. «-j = 1-6848. The second resistance, both theory and experiments show to in- crease as the square of the velocity, and from the results of De Pambour's investigations it appears that for a train of average length, moving at a velocity of ten miles per hour, this resistance amounts to about 33 lbs., which, assuming the engine to be of the most usual dimensions, is equal to '0055136 lbs. for every square foot of the piston, when moving with a velocity of one foot per minute; or, putting r for this resistance, we have rV^ when the velocity is V. As the value of r depends not only on the number and description of the carriages composing the train, but also on the size of the driving wheels, the diameter of the piston, the length of its stroke, the gauge of the railway, and even on the direction and force of the wind, it will be perceived that the value which we have assumed above is very general indeed, and that, wherever great accuracy is required, these circumstances must be taken into consideration, and the more exact value of r deduced therefrom. We have therefore, in this case, for the value of P, or the sum of the resistances against the piston. V = m(X of V, l^ a(l, + ^){n + g[{^ + rV^)(l + (?) +/+ tt + », V]} ' which being substituted in equation (XV.) for ^ (1 + ^) -f /"+ «-, aDd_^ reducing, we have for the values of V, Q, and ^a Y, as follows : — Or, substituting for n, ^, x, v and ^, the same numerical values i^b^ before, and making /*= 3*0125 lbs. per square foot of the piston'^ ' surface, when moving with a velocity of one foot per minute, whic^-^ would be its average value, these formulae become, 10000 Q ^ "■ a (6-6143 + 002753^ + -00001518 7* + 00406887)' CHAP. VII.] TH£ STEAM ENGINE. 327 Q = fQQ55(C*61*3 + -002753 e + -00001518 V« + 0040688 V); ^aV = 3632400 Q — a (-005514 7^ -f 1 -4779 72 + 2402-49 V). Condensing Engines working withotU expansion. One fonn of this class of engine is shown in Plate VII., and its general principles and mode of action have already heen descrihed at page 312. This class includes the greater numher of engines employed for propelling steam-hoats, which, although different in form, are identical in principle with the' engine shown in Plate VII. The general formulae for calculating the values of V, Q and f a V, are precisely the same as those already given at page 323, for non- condensing engines, and therefore need not he recapitulated here; but the practical formulse differ in consequence of the coefficients and constants having different numerical values. The value of 7, or the resistance on the under-side of the piston, is in this case much smaller than in the preceding, and may be taken on the average as only equal to 576 lbs. on each square foot ; the value of n and q for condensing engines will be found at page 310, and those of x, ^, and fy may be taken the same as tliose already given at paee 323, for stationary non-condensing engines. Substituting these values in the general formulae, and reducing, we obtain _ lOOOOQ ^ "■ a (2 3943 + 0030883 e) ' Y g (2-3943 -f 6030883 {) 10000 ' ^ a V = 3237280 Q — 77528 1 a V. Stationary non-condensing Engines^ working ea!pansivefy. . This engine, which is shown in Plates VIII. and IX., has been already described at page 315, and it therefore only remains to deduce the formulae by which the values of V, Q and faV may be obtained. Now we have, at page 321, and from formula (Xyil.)> by reduction, we immediately obtain 828 THB 8TKAM BNOINB. ygfn-f y[g(l+<P)-hj [part II. ^ + hyper, log ±± A And subsUtuting in these formulae the nnmerical values of the constant quantities, viz., w equal the pressure of the atmosphere, equal 2118 lbs., n and g as given for non-condensing engines, at page 810, and x and ^ as before; also, eiving to/* the value assigned to it by Comte de Pambour for this class of engine, viz., 864 lbs. per square foot of the piston's surface, we have eav = 3813883 QJr-4- + l»yP«'- log r-'"-! - 3157-73 a V; 10004 Q V = aC8-2796 + -002622 f) yg (8-2796 -f -002622 e) In order to simplify the calculation of these Quantities, and ^^r the benefit of those who are not conversant with the use of !<»£- arithms, we subjoin a table of the values of the expression /. /i + x »i -r A hyper, log j , for values of j-, (or the fraction ezpressmg tYjat portion of the stroke which is performed before the steam is cut o^j varying from one-tenth to nine-tenths of the whole stroke. Value Value of Value Value of -t ^ + hyper.log'« + ^ -'t •10 2-61258 •60 166558 •15 2-40823 •65 1 •47628 •20 2-23507 •60 1^40366 •26 208610 •66 LS1218 •80 1-95576 •70 1^26981 •33 1-87721 •76 1-20943 •85 1 •84008 •80 1-16248 •40 1-78619 •85 109862 •45 1-64194 •90 104TS9 Loeonwtive Engine working expaiuively. Lately the principle of working locomotive engines ezpannic has been successfully adopted, and we therefore give the fumf CHAP. VII.] THB STB AM BNOINB. 329 for calcolating the effects which thej will then produce. In this case we have *"'^ 7UT7) ^'--«rv'. And snhstitnting for the constant quantities the same numerical ralues as those already employed at page 326, for locomotires work- ing without expansion, we have ^ IOOOOQ(.Ji_.^.Hhyper.log^^) a (6-2993 -f -002622 ^ + 00001446 V' + -003875 y) ' V a(62993 + 002622 e + '00001446 Y* + -003875 v) 10000(^^^ + hyper.log^^) ' ^a V = 3813882 Q ( jA_ + hyper. log ^^^ ) — a (-005514 y» + 1-4779 y« + 2402-49 V). The apparent complication of these formula will disappear when we snhstitute for f j— ^ — + hyper, log / ) its value from the \ *i + ^ *i + ^ / sbove tahle, which cannot he done until the proportion of the stroke at which the steam is cut off is known. StaHanaiy Condensing Engines^ with one Cylinder working eapaneively. This enffine only differs from that already described at page 312, in its wonung expansively, which is effected by the agency of a second excentric, in the manner explained at page 315. The general formula for the values of V, Q, and e^^i ^^^ identical with those already given at pages 327 and 328, for non-condensing working expansively; but the practioJ formuln differ in 330 THE STBikM ENGINS. [PjkRT II. consequence of the constants t, w, y, ?., ^, and f^ having in this case the value assigned to them at page 327, for condensing engines working without expansion ; these heing substituted in the general formulee, they become as follows : — 10000 ^ /"-^ - i^-™ i.«'« + ^ Q = a (20953 + 0029412^) Va (2*0953 -f '0029412 6) V = — a (20953 + 0029412^) ^a V = 3399972 Q (j-^ -i- hyper, log -2-^) — 712-123aV. Condensing Engines, with two cylinders^ working expansively. This form of engine was invented by Mr. Arthur Woolfe, and patented by him in the year 1804. The peculiaritv in its mode of action consists in employing two cylinders, through both of which the same steam is made successively to pass. The details of its con- struction will be better understood by reference to Plate XII., in which fig. 1 is a section of the two cylinders and slide valves. Supposing the various parts to be in the positions shown in this figure, the steam from the boiler enters by the aperture A, and passes through the passage B into the top of the smaller cylinder C, and there, pressing on the upper surface of the piston, pro- duces a downward motion; when the piston has performed a certain portion of its stroke the communication with the boiler is closed, so that the steam acts upon the small piston only by its expansion during the remaining portion of its stroke. The rod a of the small piston, and that b of the large one, being both connected to the same beam, descend together, during which the steam which occupied the space below the small piston passes through the passages D and E into the space above the larger piston ; but, since the content of the cylinder F is larger than that of C, in doing so it expands and produces a pressure on the upper surface of the large piston, the under-side of which is in immediate communication with the condenser by means of the passage O and aperture H. As the pistons descend, the slide valves, moved by an ezcentric in the ordinary manner, change their position, and during the up-stroke of the pistons are in the situation shown in fig. 2, in which it will be seen that the steam from the boiler now presses directly on the upper side of the smaller piston, while the top of the small cylinder now communicates with the bottom of the larger one, by which the effort of the steam, in expanding from one cylinder to the other, is exerted on Uic under surface of the large piston, the upper side of the same being now in direct communication with the condenser by means of the passage £ and aperture I. The mode of action of the CHAP. VII.] THE STEAU ENGINE. 331 steam in the cylinders being then correctly understood, it is only necessary to state, that the manner of transmitting the motion of the pistons to the working point, and the arrangement of the condenser and air pump, may be the same as in other steam engines, and similar to that which we haye already described. In order to deduce formulee for calculating the effects of this class of engine, let a^ represent the area of the small piston, and a^ that of the larger piston, both in square feet; let /, equal that portion of the small piston's stroke which is performed before the steam is cut off, l^ the whole length of the small piston's stroke, and l^ the length of the larger piston's stroke ; also, let p ^ be the pressure of the steam in the small cylinder before the expansion commences, Pa the pressure of the same in the large cylinder before expansion, and ^, and A^ the length of a portion of each cylinder, equivalent in content to the space left for clearance and the steam passages. Then we have from equation (XI.*), page 319, for the work down in the small cylinder, «i (^ -rp, ) 1^ -f (/x + Xi) hyper, log ^^^^ - ^«i U- Now, in order to obtain the work done by the expansion of the steam in the large cylinder, put «, = a, /^ = the space moved through by the small piston, «, =0^/3 = the space moved through by the large piston, and « ^ a, / = Vie space moved through by the same at any portion (/) of its stroke. Then, the space occupied by the steam (having a pressure equal to p^) before the commencement of the stroke, equals 9^ -f^i^i -f-^^Xg, and that which it occupies at any portion (/} of the strolce equals «-fa,X| -{• a^x^; therefore, patting |3 = ajAj H-a^Ag, we have from formula (IX.), for the preasare of the steam due to its altered volume, n \ Then, since A« represents the elementary space moved through by the larger piston, we have for the work developed during the motion, pAty which, by substituting the above value of/?, becomes ^. As /n \ n therefore, the whole amount of work developed is represented by ('.+^)J:i-^(- +/»,}-- J:a.; which being taken between the proper limits of « = «| , and « = «j| ' equals <"*«/:'.-Ti(=+'.)-5/:-^ 332 THE 8TBAM BNOINB. [PABT II. whence, by integnting, we obtain (,, +^)hyper.log^l±| (J +/>t ) - ^(*, "'i)- Then, in order to eliminate />,, we can derive its value from equation (IX.) in terms ofp^ , viz., which being substituted for /?, in the expression above, it becomes and again, substituting for «^, x^, and 0, their several values above, and reducing, we have -x(^ +^)(- +/'.) hyper, log ^-l^^-^:^^!^^^ for the whole amount of the work done by the expansion of the steam in the large cylinder, minus the resistance produced by the reaction of the steam in the small cylinder ; whence, adding this to the expression already obtained for the work done in the small cylinder, and reducing, we obtain . -f hyper, log ^^f/-^^-J;-^"-^n)--^a,/3, which equals the actual work performed by Woolfe's engine, durin^^ "I one stroke. Then, in order to obtain the value of P, in this case the resistances^ against each square foot of the lar^r piston's surface, arising fron^^* the load and friction of the engine, and from the imperfect vacuun^^^ in the condenser ; let w, /, and ^, represent the same quantities a ^^^ before, only taken per unit of the large piston's surface; thei— *^ we have P = ^(l +^)+/+^, which being multiplied by a, /,, equals the whole amount of rcr " sistance developed during one stroke of the engine ; equal Therefore, since where the engine has attained uniform motion the resistance must equal the work performed, we have Q- CHAP. VII.] THB nUU XNGIMB. 833 ■h hyper, log ^^^^;^-^\>-^^^^n)-%,/3 Then, if V equals the yelodty of the larger piston, and 9 the V nnmber of strokes which it makes per minute, we have » = -j- , *s whence we obtain, for the value of j^^, from equation (XIV.) * «iyV(/, +Xi) g' Then, snbsdtuting this valae of />, in the above expression, rednciug, and solving in respect of Q, V, and ; a, V, we have «.Vf» + <rr> (! + »)+/+>]? -ll- + hyper, log ^±-i. + hyp«. log •«J''Jt-*«>-+-"l^J Q {^ + Hyper. U^ ;4il + hypT. U«^f±^4±fJ} -«.v £. i + f In order to put these formulsB under a more practical form, we must snhstitute for the constants and coefficients their average values, vis., for n, 7, and ar,as before, for condensing engines, /*» 125 lbs., 9 as before, x, = 'OS/j, and x, = '05/,, then we obtfun ie>«0Q{^24-H hyper. log;^^hyper.U.^4,^^±S^} ^ " «, {2-231S + •0029412 {) (•,V-S9D007SQ {•9624 + hyper, log ^±^' 334 THB STBAM RNOINB. [PART II. The following table, giving the numerical value of the expression hyper, log i±^+h.yper.logji^±^\±^) for jslues of -p varying from 77; to - , and for the three cases where — eqaals 4, 3 and 2, may sometimes be found useful : in this table the two strokes are assumed to be of the same length, that is, /^ == Z^. . ( Value of Value of (hyper, log ^ + hyper. lo| «s((i + ^)+M,\ ^«i(/, + ^.) + W when a, « 4a). when a, « 3a|. when a, = 2a,. •10 316969 2-92674 2-57162 •15 2^88200 2-63906 2*28393 •20 2-65886 2-41591 206079 •25 2-47654 2-23359 1-87847 •30 2-32239 207944 1 72432 33 2-23901 1-99606 1 64094 35 2-18885 1-94591 159078 •40 2-07107 1-82813 1-47300 •45 1-96561 1-72277 1-36764 •50 1-87088 1-62793 1-27281 Sin^e-acting Engine^ in which the Steam cuUa only upon the upper surface of the Piston. This class of steam engine is that usually known by the appella- tion of the Cornish pumping engine, for which purpose it is more peculiarly adapted ; and from the surprising amount of work which has been realized by means of it, with the consumption of a given weight of fuel, has attracted a great deal of attention, and has become almost universally adopted in mining districts. As it is not our province here to enter into a detail of tbe mechanical construction of these engines, we have only shown in plate XII, figure 3, a section of the cylinder and valves ; it maj, however, be remarked, that these engines exhibit the greatest me- chanical skill in their design and construction, and doubtless ooe cause of the high duty which they perform results from the accuiacj and precision with which the various parts are formed, and the precautions adopted for preventing loss of heat by the steam io passing through the engine. In the plate, A is the cvlinder, which is surrounded by an outer casing sufficiently large to leave a space of about an inch round the whole of the exterior of the cylinder, which space (termed the steam jacket) is always kept filled with steam of the same pressure as thtt m the boiler, by means of the pipe B, which is in direct communici- CHAP. YII.] THB STEAM ENOINB. 335 tioD with the hoiler ; a similar space is also left helow the cylinder, and kept constantly filled with steam. The ohject of thus sur- rounding the cylinder with hot steam, is to prevent its interior surface from heing cooled during the up-stroke; and for the same purpose, the top of the cylinder, as also the upper part of the valve box, and the entire steam pipe from the boiler to the cylinder, are enveloped in an external case, the space between being filled with sawdust or some other non-conducting substance. The piston being at the top of the cylinder, the valve C is opened^ allowing the steam to pass from the boiler into the cylinder, where, pressing on the upper surface of the piston, it causes its descent, raising in so doing the pump rods and counterweight which are attached to the other end of the engine beam. After the engine has performed a portion of its stroke (varying from -^^ to ^ in different engines), the valve C is closed, and the remainder of the downward, or, as it is termed in Cornwall, the in-door stroke, is performed by the expansion of the steam : the under -side of the piston is in communication with the condenser by means of the pipe D, which is closed by the valve E towards the conclusion of the down-stroke. There being in the Cornish engine no crank, or other means by which the precise length of the stroke is determined, it becomes necessary to adopt some means of preventing the piston descending too far, and by doing so injuring the bottom of the cylinder ; the way in which this is effected, is by so regulating the quantity of steam admitted through the valve C at the commencement of the stroke, that it shall be just sufficient to carry the piston to the proper distance, and no further; but since any variation in the pressure of the steam in the boiler, or in ihe quantity of water raised by the pump, would immediately destroy this adjustment, and cause the piston either to fall short of or to exceed its proper stroke, springs are placed so as to receive and stop the beam of the engine when it descends too low, and at the same time a bell is rung, which warns the engine-driver to lessen the quantity of steam admitted by the valve C. The piston having been brought to a state of rest at the bottom of the cylinder, the valve F is closed bv which the com- munication with the condenser is cut off, and the valve G, termed the equUihrium vaive^ is opened, by which a free communication is effected let ween the top and bottom of the cylinder, and an equilibrium esta- blished between the pressure on the upper and under side of the piston, which, therefore, having now nothing to oppose its motion but the friction of the engine, is drawn to the top of the cylinder by the connt^^eight which it had raised during its down-stroke, and which is sufficiently heavy to raise the columns of water in the various pmnp mains, and to overcome the friction of the engine. As, how- ever, the piston would continue its motion until it came into contact with the cylinder cover, were it not checked, to do which the equi- librium valve G is closed before the piston has completed its up- stroke, after which the further motion of the piston, by compressing the steam on its upper side and attenuating that below it, occasions a 386 THB ST£AM BKOINB. [PABT II. Bofficient preponderance of pressure on its upper surface to bring it gradually to a state of rest. The piston is now ready to resume its downward stroke as soon as the valve C shall be opened, and the steam from the boiler admitted to the top of the cylinder; this is effected at the proper time by means of a contrivance termed the cataract, by which the time which elapses between the up and down strokes can be regulated at the pleasure of the person who baa charge of the engine. The various contrivances by which the valves are opened are not shown in the plate, as it would require too long a description to render their use clearly intelligible ; it is sufficient to remark, that the valve C can be closed at any portion of the stroke which is desired, by means of a simple adjustment ; that the exhaus- tion valve F is so arranged as to close somewhat before the equi- librium valve G is opened ; that the injection cock of the condenser is only open during the same time that the exhaustion valve F is open ; and that the valve F is opened somewhat before the valve C, in order that the steam beneath the piston may be perfectly con- densed as soon as the down -stroke of the engine commences. We will now proceed to deduce the formula for calculating the effects of these engines, but since the nature of the work performed by the Cornish engine during its in-door and out-door strokes are so essentially different, it will be necessary to investigate each separately. First, then, during the in-door or down stroke, we have the pres- sure of the steam direct from the boiler upon the upper surface of the piston during the first portion of its stroke, which pressure will be so little less than that in the boiler itself, that without any appreci- able error it may be assumed to be the same, for the piston, engine beam, pump rods, and counterweight, comprising a joint mass weigh- ing several tons, require considerable force to put them in motion, and only acquire that motion very gradually, so that the steam has time to attam in the cylinder the same density and pressure as it had in the boiler; and during the after part of the stroke, we have the force due to the expansion of the steam above the piston. Now, if we put a for the area of the piston, x for the length of the cylmder equivalent to the clearance ana steam passages, p. for the pressure- in the boiler, l^ for the portion of the stroke pertormed before ex- pansion, and l^ for the whole length of the stroke from its com- mencement, we have from formula (XL*) for the whole amount of work performed during the in-door stroke of the engine; or, indeed, during both the in-door and out-door strokes, for the latter is, as already explained, performed by the descent of a counterweight which was raised during the in-door stroke ; and since the work expended in raising it is precisely equiva- lent to that which it performs during the out-door stroke, the amount which represents it would have to i^pear first on one ode of CHAP. VII.] THE 8T£AM BNOINB. 387 the equation as a resistance, and afterwards upon the other as work performed, we shall therefore simplify the calculation hy omitting it altogether. Now, the actual resistances to which the Cornish engine is ex- posed are as follows; viz., during the in-door stroke* the pressure on the under side of the piston arising from the imperfect vacuum, which we will as hefore designate hy v; the friction proper to the motion of the engine when unloaded, equal /*; and the additional friction caused hy the load of the engine, represented by ^ ^j during the down stroke, and ^^^ during the up stroke, ^ having a different value in consequence of the different load upon the engine, and ^ being the effective load measured per square foot of the piston's surface: there is also a slight resistance occasioned by the steam pressing with rather more force upon the upper surface of the piston than the lower, in consequence of its having to pass through the passage and equilibrium valve, but this difference of pressure is too small to re- quire being taken notice of; the compression of the steam, however, in the upper part of the cylinder at the conclusion of the out-door stroke, and after the equilibrium valve has been closed, is of more consequence, and requires to be considered. If, then, we put p^ for the pressure of the steam on each side of the piston at the moment that the equilibrium valve is closed, p^ for the pressure of the steam above the piston at the conclusion of the out-door stroke, and /, the length of the stroke performed after the valve is closed^ we have from equation (IX.), or, sabstitnting for />,, in the last equation, its value from the first. {,^(^-)}-7 Then, if / represent the length of the piston's out-door stroke remaining to be performed at any moment, and p the pressure of the steam above the piston at the same moment, we have from equation (IX.), tnd for the work performed in the elementary space A/, we have apAly in which, substituting the above value of /?, and taking the integral between the limits of / ss 0> and / =: Z,, we have z 338 THE 8TBA1I BNOINE. [PABT II. p^ dl in \ n /-•» = ax(^ +>'0 ^^^'' ^^^^1( 7*'-' which hccomee, hy Bubstitating for p^ its value above in terms of/?,, which expression equals the work required to be done by the piston in compressing the steam in the upper part of the cylinder, after the equilibrium valve is closed. But a portion of this work is perfonned by the expansion of the steam on the under side of the piston, and therefore we must deduct this from the above. Now, the pressure of the steam on the under side of the piston at any moment being represented by />, and the length of the piston's stroke from its commencement at the same moment b^ng represented by I, we have and for the work performed in the elementary portion of the stroke represented by A /, we have ap A /, in which, substituting the abov& value of py and integrating between the limits of I z= l^^ and^ / = /j "— ^3> we obtain r substituting for p^ its value above, a(/.-^,H-X),fJ^,(-; 4-;>.)hyper. log ^±^±-+1.!^ equal the work which the steam below the piston performs by ^l_ ts expansion after the equilibrium valve is closed; and therefore t^^fc® resistance actually to be overcome by the engine equals i CHJU>. VII.] THB 8TXAM BNOINB. 889 Therefore, the total resistance during both strokes of the piston, equals 2Pa/, = a4 {e(l + (p, 4. (p,) + T + 2/) - (/, - /« + A) hyper, log ^-i^-i-^ J ; and therefore, from the necessary equality between the work and the renstance, we have = a/,{e(l +*, + «),) + »+«/} -(/,-/3 + X) hyper. Iog^A+i_J. Now, since the Cornish engine only performs work in pumping during its in-door stroke, it is usual to take its effective Telocity as y/g = V, v being the number of double strokes which the engine mi^es per minute. Then, the quantity of steam actually consumed by the engine each double stroke, is only that which remains below the piston when the equilibrium valve is closed, and which equals a (/, — /, + X) ; and further, since the quantity of steam generated eqoak that consumed, we have 1 V m wbich v, = , and y = - ; making which substitutions and solving in respect ofp^^ we obtain wUeh latter expression is the value of j^,, alreadv given at page 887. Then reducing, and solving in respect otp^^ we have _ /,Q /, -i-2X n ^»-a^V(/,-/3 + X)- /, + x 7- z 2 340 THB 8TBAM BNOINB. [PABT II. to a '3 a '^ 2 -I a ► 'So I 08 O If- •I? •SIS p CO '■s 2 •a 8l« + + I: + + > ^ + II Of ? I: I: + I I + ^ /< /< 04 04 + + + 04 •^ *J^ ^* Is + 9 /< /< X + /< + /< •f 1 1 1 -**• t t -^« o 2 o ^ p i ,£ 3 j: 3 II g^ + I s 04 + Is > I + or H > CRAP. VII.] THE 8TBAM BNOINE. 341 Single-acting Engine^ in which the steam presses only upon the lower surface of the piston. The engines belonging to this class are better known under the title of atmospheric engines, from the down stroke being performed entirely by the pressure of the atmosphere upon the upper surface of the piston. Plate XII, figure 4, is a section of the working cylinder of an atmospheric engine ; the steam at a low pressure enters the bottom of the cylinder A, by the valve B, and assists the counterweight at the other end of the beam in raising the piston. When a certain portion of the stroke has been completed, the valve B is closed, and the counterweight being insufficient of itself to overcome all the resistances of the engine, its velocity becomes gradually diminished, and the piston at last comes to rest at the top of the cylinder. The valve C is then opened, by which a communication is effected with the condenser; and a vacuum being formed under the piston, the pressure of the atmosphere upon its upper surface causes it to descend. In order to prevent the piston making too long a down stroke and coming in contact with the bottom of the cylinder, the valve C is closed some time before the conclusion of the stroke, and the oncondensed steam then remaining in the cylinder being com- pressed below the piston, gradually brings it to a state of rest. The work performed by the steam during the up stroke will be represented by formula (XL*) and if we put n for the pressure of the atmosphere, that performed during the down stroke will be lla/g, therefore we have « (^ + Pi) {^1 + (^1 + X) hyper, log ^±^^ - %/, + na/^ for the whole work performed during both strokes of the engine. Then we have for the resistance resulting from the friction of the engine when moving without any load, 2/2/9 and for the additional friction caused by the load, together with the resistance of the load itself, /^eO +?i + ^s)' ^^^9 finally, for the resistance caused by the imperfect vacuum, if we put l^ for the length traversed by the ]nston after the valve C is closed, we have for the resistance before the closing of the valve, a n- (/j — l^) ; then if we put p for the pressure of the uncondensed steam, at any moment when the piston 18 at a distance equal to / from the end of its stroke, we have from equation (IX.), l-\-\\q ' "J q whenee the work to be performed will be represented by ^_ /3-f X /n \ n •<'.-<fTr(i-)-"l/^' = a(/,+X)(^ + ir) byper. log ^±^_2a/,. « (^3 + ^) f - + » j hyper, log 342 THB BTBAM ENGINE. [p Therefore, collecting these several resistances, we have T w I uypcr. lUK — -la/, +a^,{f(l+^, +f,) + a* (/,-/,) = a ( ^ + jP. ) |/, + (/, +'X) hyper, log ^ Now, in order to find the quantity of steam consnmed engine, we most deduct the quantity left below the piston w! yalVe C is closed, from the quantity admitted through the ' during the early portion of the stroke; the latter equals a{ with Uie pressure p^, and the former a (/^ -|- X) with the prea then, in order to find the space (equal S) which it would oc tha pressure />|, we have from equation (IX.) and solTing in respect of 8^ we have B = a(/, + X)!L±i^. And therefore the quantity consumed per minute equals ,a{(/.+X)-(/,-.X)l±i^} = Q,.; then, substituting for v and v^ their yalues as before, redact solving in respect of />j, we have '^'~«yv(/. + x)^/j-|.x\,, ^ '; ?' Then, substituting this value of p^ in the preceding ei reducing, and solving in respect of V, Q, and f a V, we obtain GHAP. Vll.] THE STEAM ENGINE. 343 V- <<.v- •"*'.+«{'-'-¥(=+')[,Ti-.+''— -mi! ' Tbe remark made above with regard to the Cornish engine, of the want of experiments from which to deduce the values of the several resistances, will equally apply to tbe atmospheric engine, and we are therefore obliged to leave the above formula in their general form. PRACTICAL APPLICATION OF THE FOREOOINO FORMULA. In the preceding investigation we have deduced for each engine three forms of expression, giving the velocity of the piston in feet per minnte or V, the quantity of water in cubic feet evaporated by the boiler per minute or Q, and the effective useful work performed by the engine per minute expressed in pounds raised through a height of one foot or ^aV ; it may, however, be desirable, before concluding the subject, to deduce ^om these, expressions for the horse power, and t¥e quantity of coals consumed. In estimating the horse power of a steam engine, it is usual to assume the power of a horse to be equivalent to raising 33,000 lbs. through a height of one foot in. a minute; and, therefore, to derive the horse power of an engine whose effective useful work per minute equals ^aV, we have ^gy _^ J the number of horses' power 33000 ""I of the engine. Then, if we put F for the weight in pounds of coal, which is required to evaporate a cubic foot of water, we have ^ r the number of pounds of coal consumed ^ I by the engine per minute, QF J the number of pounds of coal consumed and 33000 — - = | p^^ horse power. The following table contains the value of F for four different varieties of coal, compared together by Mr, Wicksleed. 344 THB STEAM BKOINB. [part II- Description of Coal. Na of Ibf . of Coal required to evaporate one cubic foot of water «» F. Welsh coal, used in Corawall . . Best ditto, used at Old Ford . . . Anthracite, ditto . . . Best Newcastle small, ditto . , . Derbysliire coals, ditto . . . 6-7414 6-5838 6-9337 7-3322 9-2292 TABLE I.— Quantity of Coals equivalent to the horse power or 33,000 lbs. raised one foot per minute in high pressure steam engines, when the greatest possible effect is obtained *. 2S4 5 251 275 21*2-8 307 7 3202 343*1 Is 45 60 m 180 210 ^11 II- 74 U'B 29-7 44-5 142 104 Quant itr *nf maJ equfrAksi ta ooe bone potter* Foun<ls rai&fil otic fopi hi|^h «^i^- r Aleut to th^ Immediate pover of When working ■I full pivHiire,. 480 1*>3 08 82 74 70 6S 143 77 &0 51 48 414 mt fulL pi^MUFe. llB. 2,70a,ooo 8,200,000 13,700,000 ie;,@oo,ow 18,000,000 10,200,000 20,BOO,OQO Wbeti «oTlU.fi|| 9,30a,000 17,700,000 33,7041,000 26,2i>O,OO0 2g,704i,a(ia 32,200,000 k TABLE IL — Quantity of Coals equivalent to the horse power om 33,000 lbs, raised one foot per minute in condensing steam engines^ when the greatest possible effect is obtained. t c^ ll^ PcKiQdf raiwd one fofil hift* egml- ^ s. g ^M s Qtiant ity of €Oft] eaaivaJcnt to ¥sk9tt to thfr tUttwft'Btr paver (x^ ^mcr^ fl ll £ *B. <iOm hone power. thw cteim produ Efil »ct by 8« lli^ OM ^ Mif tL Wh*n acbog 1 S worn warning ^ §'3 9.% full fireMUre* WhffD iForiLlnia H Eipuultely^. Bt Ml proiun. npuulrtJjt i- II1C}1I?I. ItM. Itii. Ibt. Ibi. Ibi. 2^0 35 2-5 63| 40i 2l,<KM3,0OO 33,100^000 "* J 2^4-5 45 7-4 63 38* 21,400,000 35,200,000 2fil'0 00 1 14-8 60 35! 22,400,000 37.500,000 ^ / 275i) »0 1 207 331 40.000,000 / 292 8 120 445 32i 41,000,000 / 307 7 150^ 1 593 32 42,400,000 / a^2 ISO 74 2 314 42,700,000 / 3436 240 104 at 43,500,000 J * The tables here given, marked I. II. III., were extracted, with the av thor*! peroiisrion, ^m Mr. Tredgold*8 work on Railroads. CHAP. Vll-I THE 8TB1M ENOINB. 345 Remarhon Tobies I. and II. — The colamns showing the pounds in engme oug^t to raise one foot high, by the heat of one bnshel of coals, are added chiefly for the pnrpose of comparison with actual practice. Now, it is stated, that after the most impartial examina- tion for seTeral years in succession, it is found that Woolf's engine, at Wheal Abraham Mine, raised 44,000,000 pounds of water, one foot high, witb a bushel of coals. And, ^^ the burning of one bushel of good Newcastle or Swansea coals in Mr. Watt's reciprocating engines, working more or less expansively, was found, by the aeconnts kept at tbe Cornish mines, to raise from 24,000,000 to 32,000,000 pounds of water one foot high; the greater or less effect depending upon the state of the engine, its size, and rate of working, and tbe quality of the coal." We shall further add the results of half a year's reports taken, without selection, from Lean's Monthly Reports on the work per- fonned by the steam engines in Cornwall, with each bushel of coals. The numbers show the pounds of water raised one foot high with etch bushel, from January to June, 1818. Name of Pounds raised one foot by a bushel of coals. January. February. March. April. May. June. Stotf ComnMo Bb- |WW VOT <Woolf% ^WtedAbnlMm (ditto) .Ditto (ditto) ■Whad Unity (ditto) 'Daloomh Cneine WlMl Abraham Engine iUiilsd Mines Cngj iTicskiiby Encine [WhsalCfaScgEn Engine >J8B,000 ao.834,000 41JM7.000 314X10.000 42^89,000 aa,sw,ooo 38.306,000 38,733.000 98,406,000 SS.4S4,000 26,108.000 3S.364,000 98,000.000 393)6.000 41.364.000 36,180,000 31.830,000 30,375.000 38,319.000 91,808,000 29,969.000 99,611,000 26,064.000 23,606.000 23,836.000 30.445,000 96.978,000 40.490.000 35,715.000 31.427.000 41.867.000 33.594.000 39.723.000 23.e86.r" ,000 29, 29,032.000 30.336.000 31 .520,000 '34.352,000 1,702.000 '34.846.000 41.888.000 33,934,(NN) 33.564,000 41.823.000 33,932,000 38.233.000 '38.143.000 33.714.000 ,34.291 .0(N» 33.967.000 '30.105.000 40.615.000 '42,098.<NIO — 35,797.000 Tliese numbers are less than the immediate power of the engines, bj the friction and loss of effect in working the pumps ; hence, in comparinff them with Mr. Tredgold's table, it may be inferred that be msde bis calculations from such data as can be realised in practice. It is known from experience, that a cubic foot of water can be con- verted into steam equal in force to the atmosphere, with 7 lbs. of Newcartle coals; but we also know the attention necessary to pro- dace that effect, and therefore have assumed that %-f^ lbs. will be wjoired for that purpose. Aeeording to Mr. Leans Monthly Report, for January, 1 833, the foDowing engines raised more than 50,000,000 pounds, one foot * r consuming % bushel of coals : — 846 THE 8TBAM SNOINB. [PABT II. Name of Mine. Diameter of cylinder. Pounds raised bya bushdofoMls. Strokes Cardrew Downs . . . Binner Downs . . . . Ditto Consolidated Mines . . Ditto Ditto Ditto Ditto Ditto Polgooth Pembroke East Crinnis Wheal Leisure . . . . Wheal Vor Ditto Ditto Poladras Downs . . . Great Work Wheal Towan . . . . Ditto Wheal Falmouth Consols Wheal Darlington . . . 66 inches 70 65 90 70 65 90 90 65 66 40 76 66 80 80 53 70 60 80 80 70 80 single 51,831,751 57,942,435 55,931,852 51,713,913 61,846,133 54,726,957 59,978,983 52,040,672 65,617,011 70,240,452 61,170,237 62,097,533 53,506,372 88,504,900 65,471,147 53,938,177 56,766,668 65,460,248 73,159,628 68,782,390 54,334,137 66,058,518 734 1013 8-50 4-86 5-93 8*60 4*69 6-50 5-28 800 6-67 704 358 6-22 607 6-37 8-78 608 4-90 6-98 6-22 8-47 Of the above, the engine of greatest operation, the first at Whesl Vor, raises the water 190 fathoms, ^i seven lifts, drawing perpen- dicularly 160 fathoms, and the remainder diagonally. Main beam over the cylinder ; stroke in the cylinder ten feet ; one balance-bob at the surface, and three under ground. The following extract from Leans Monthly Report, for June 1841, will show the advance which had been then made. Name of Mine and Engine. Wheal Darlington, Eastern . . Great Wheal Fortune, Wheal 1 Prosper ........ J Duffield WhealJulia. . . . Godolphin, Sims* . . Great Work, Leeds* . Wheal Vor, Borlase's . Ditto, Trelawny*s North Roskoar, New . Wheal Unity Wood, WUliams's Hallen-Beagle Diameter of cylinder, in indies. 80, single 80 80 80 60 80 80 70 80 70 Poundi lifted one foot bra bushel of ooala. No.ofsti6kci perminut& 81,681,776 9-76 64,169,466 6-46 64,438,341 6-68 60,966,983 6-77 60,166,186 630 66,776,208 6-70 74,927,176 6-90 60,634,127 6KK> 67,364,238 4-76 61,168,649 6-64 62,314,765 6-60 CHAP. VII.] THB STBAJi BNOINB. 347 Nun* oi Mine «iid Ea^am. D1uD«leror Pftundi lifted Ddv foot by » buBbti<frcc*Ift. ^fo.flflt^a1tM pcrmLtiiitew Wheal UnitT 70, tingle 7a M 50, combined 85, kitig]« m> „ 30 „ B5 „ 85 „ 50- .. ao „ 80 „ 66 „ 08,462,34 fi 70,(J35,787 88,096,178 9«,659,570 55,219,358 61,652,653 €7.044, 127 a,i3i,7ei 54,477,451 59,2tf7,244 77,8M,927 71,144,002 6-70 3-60 4-30 4-00 B-40 1030 7i)7 660 3-65 436 530 5W Catq Br«ft . , , . . INtto, Siou' . * * United Miiiei, Tajlori' * DitUi, OurdoEo'i . Ditto, Elfbn's , Ditto, Loam'i . , Ditto, Hocking'i United HilJif WiUiami^f . Fowef CcHiaolfj Aa»tin*f . PfllirMitll TABLE in.— /S^oiriny r^ gfecto of a force of traction of 100 Ibt. at different velocities^ on Canals^ Railroads^ and Turnpike-roads *. {From Tredgold,) TdodlTOflfcieloA. Lo«d nwjvtd by « power of 100 llw- •HBIlll. On » C»a»l- On » l»Td lUiliny, Onftln«l TuTDpDw Roadi mus DTMved, Total ErH>vddt tllfert. Tot*I IDAtl r 5 6 1 8 10 13-5 366 4-40 5-13 1 6-86 7-33 8^80 ' 10-26 11 -73 13-20 14-66 10-0 Ibi. 55,500 38,542 Si8,310 21,060 ,13,875 9,635 7,080 5,420 3,468 1,900 Ibt. 39,400 27,361 20,100 15,890 9,850 6,840 5,026 3,848 3,040 2,462 1,350 Itat li,400 14,400 14,400 14,400 14,400 14,400 14,400 14,400 14,400 14,400 14,400 lb*. 10,800 10,800 10,800 10,800 10,800 10,800 10,800 10,800 10,800 10,800 10,800 1,800 1,800 1,600 1,800 1,800 1,800 1,800 1,800 1,800 1,800 U800 1,350 1,350 1,350 1,350 1,350 1,3150 1,360 1,350 1,360 1,350 1,350 This table is intended to exhibit the work that may be performed by the same mechanical power, at different velocities, on canals, rail- roads,' and tampike-roads. Ascending and descending bj locks or canals^ may be considered equivalent to the ascent and descent of in- clinations on railroads and tampike-roads. The load carried, added to the weight of the vessel or carriage which contains it, forms the total • Though the force of traction on a canal varies as the square of the velocity, the me c h anical power necessary to move the boat is usually reckoned to increase as the cube of the velocity. On a railroad, or turnpike, the force of traction is constant; but the mechanical power necessary to move the carriage inereases as the vdodty. 348 8TRBN0TH OF ANIMALS. [PART II. mass moved ; and the useful effect is the load. To find the effect on •canals at different velocities, the effect of the given power at one velocity being known, it will be as 3' : 2*5^ : : 55,500 : 38,542. The mass moved being very nearly inversely as the square of the velocity ; at least, within certain limits. This table shows, that when the velocity is five miles per hour, it requires less power to obtain the same eflfect on a railway than on a eanal ; and the lower range of figures is added to show the velocity at which the effect on a canal is only equal to that on a turnpike- road. By comparing the power and tonnage of steam vessels, it will be found that the rate of decrease of power by increase of velocity, is not very distant from the truth ; but we know that in a narrow canal the resistance increases in a more rapid ratio than as the square of the velocity *, that is, within certain limits ; beyond them, there is a remarkable change in the circumstances of resistance. Sect. IV. Animal Strength as a Mechanical Agent. 1. The force obtained through the medium of animal agency, evi- dently varies, not only in different species of animals, but also in dif- ferent individuals. And this variation depends, first, on the particular constitution of the individual, and upon the complication of causes which may influence it ; secondly, upon the particular dexterity ac- quired by habit. It is plain, that such a variation cannot be sub- jected to any law, and that there is no expedient to which we can have recourse but that of seeking mean results. Secondly, the force varies according to the nature of the labour. Different muscles are brought into action in different gestures and positions of an animal which labours ; the weight itself of the ani- mal machine is an aid in some kinds of labour, and a disadvantage in others ; whence it is not surprising that the force exerted is different, in different kinds of work. Thus the force exerted by a man is * Aooording to the interestinff researches of Du Baat, the retistaDoe to the motion of boats, even in canals, may be regarded as proportional to the square of the velocity, or R as V* neariy, provided R be made to depend upon the transverse sections of the vessel and the canal in which it movea. If c be the vertical section of the canal, and b the vertical section of the immersed por. tion of the boat, or barge ; then The mean of Dn Buat'i ezperimmiu givM, K = 8'46, or ^ - 8-46 ' but these experiments were not so numerous and varied as might be wished. See PrincipeM d* HpdrauHque^ tom ii. pp. 340, 342, &c. CHAP. VII.] 8TRBNGTH OF ANIMALS. 349 different, in carrying a weight, in drawing or pushing it horizontally, and in drawing or pushing it vertically. Thirdly, the force varies according to the duration of the lahour. The force, for example, which a man can exert in an effort of a few instants, is different from that which he can maintain equahly in a course of action continued, or interrupted only hy short intervals, for a whole day of lahour, without inducing excessive fatigue. The former of these may he called Absolute Force j the latter Permanent Force, It is of use to hecome acquainted with them hoth, as it is often advantageous to avail ourselves sometimes of the one, some- times of the other. Lastly, the force varies according to the different degrees of velo- city widi which the animal, in the act of labouring, moves either its whole body, or that part of it which operates. The force of the animal is the greatest when it stands still ; and becomes weaker as it moves forward, in proportion to its speed ; the animal acquiring, at last, such a degree of velocity as renders it incapable of exerting any force. 2. Let ^ be a weight equivalent to the force which a man can exert, standmg still : and let V be the velocity with which, if he pro- ceeds, he is no longer capable of exerting any force : also, let F be a weight equivalent to the force which he exerts, when he proceeds, equably, with a velocity v. Then F will be a functiou of v, such that, 1st, it decreases whilst V increases ; 2nd]y, when » = 0, then F = ^ ; 3rdly, when v = V, F = 0. 3. Upon the nature of this function, we have the three following suppositions. 1. Fs^fl— -V (Bouguer, Man. dee Vais.) 2. Feaf^i—ljY (Euler, Nov. Comm. Pet. tom. IIL) 3. Frs^Tl— M. (lb. tom. VIII ; and Act of Bowers.) 4. CoroH. 1 . The effect of the permanent force being measured by the product F v, the expression for the effect will be one of the three following, accordingly as one or other of the suppositions is adopted. 1. Fv(l-|),or^«(l-|.). ^- ^^ a/('- !)'"*"('- S)- 3. Pv(l.-^!),or^. (l-^y. 350 ANIMAL 8TRBNQTH : SCHULZB's BZPBRIMBNT8. [PART II. 5. Coroil. 2. To know the weigbt with which a man shoold be loaded, or the velocity with which he onght to moTe, in order to pro- dace the greatest effect, we most make ^ . Fo e=s 0. Whence we shall haye 1. F = - ^; and v = 2 2. F = - ^ ; and v = 3. 4 >/3 and I? = - V. 3 V = 0-5773 V. 6. CoroB, 3. And the yalae of the greatest effect, according to the seyeral hypotheses, will be. 1. \^y: 2. 3 >/3 f V s 0-3836 f V : 3. 27*^- Bat which of the three suppositions ought we to prefer ? And sre we certain that any of them approximates to the true law of nature? Mr. Schuize made a series of experiments with a view to the de- termination of this point*, and with regard to men decided in favour of the last of Euler s formulae : viz. ,=,(.- 1)-. As the experiments of this philosopher are very little known in England, I shall here present hss brief account of them. 7. To make the experiments on human strength, he took at rao* dom 20 men of different sizes and constitutions, whom he measured and weighed. The resnlt is exhibited in the following table. Order. Height. Weight. Order. Height Weight 1 5/ 3// 4W 122 11 5/ 9// 7/// 132 2 6 2 3 134 12 6 1 4 157 8 6 7 2 165 13 6 8 2 175 4 6 6 181 14 5 4 1 117 5 6 11 2 177 16 5 10 8 192 6 6 4 158 16 6 8 133 7 6 8 3 180 17 4 11 2 147 8 6 2 1 117 18 6 8 9 124 9 6 4 8 140 19 5 6 163 10 6 4 126 20 6 10 1 181 * Mem. Acad. Sdcnc. Bsrlin, for 1783. V^ll.] ANIMAL araKNOTH: SCHULZB's BXPBRIMENT8. 851 e the heights are expressed in feet (marked ^), inches C')y m ("')y the feet being those of Rhinland, each 12-35 English The weights are in pounds, which are to our ayoirdupois 30 to 29. ind the strength that each of these men might exert to raise a yerticaily, Mr. Schnlze made the following experiments : — took Tarious weights increasing bj 10 lbs. from 150 lbs. up to B. ; all these weights were of lead, having circular and equal To use them with suodess in the proposed experiments, he the same time a kind of bench made, in the middle of which hole of the same size as the base of the weights : this hole was y a circular cover when pressed against the bench ; at other t was kept at about the distance of a foot and a half above the bj means of a spring and some iron bars. To prevent the . with which this cover was loaded during the experiment from ; down the cover, lower than the level of the surface of the he had several grooves made in the four iron bars, which sus- the cover, and which at the same time served to hold up the at any height where it might arrive by the pressure of the I as soon as the pressure of the weight ceased. Mr having laid the 150 lbs. weight on the cover, and the other s in succession increasing by 10 lbs. up to 250 lbs., he made [lowing experiments with the men whose size and weight are ibove, by making them lift up the wights as vertically as pos- II at once, and by observing the height to which they were i lift them. The annexed table gives the heights observed for Fereat weights mariced at its head. BO IGO 170 180 194 200 iio 223U 230 240 MO m *f m it m ri m jtf m U Hi it tit ** H* if aif'i ff Iff B 4 4 11 4 4 a a 2 B 1 1 I 10 6 6 5 7 4 7 3 u 2 5 6 17 3 7 3 6 5 a 9 4 11 4 3 , 3 8 3 1 1 4 7 6' 7 2 d 10 5 3 4 7 4 4 7 3 2 1 3 M 1 9 7 8 5 7 10 7 1 5 10 6 6 4 1 1 U 13 5 12 8 n 10 I 8 a 3 8 I 11 2 11 11 3 10 5 9 3 8 1 6 9 & 3 6 J 3 2 1 10 s 4 8 11 a 1 « 11 h SO 8 3 7 1 5 fl 4 I 3 1 3 « b 4 7 3 g 2 5 1 7 4 B table proves that the size of the men employed to raise the ts Terdcally has considerable influence on the height to which ronght the same weight. We find also that the height dimi- in a much more considerable ratio than the weight increases ; e may therefore conclude, that it is advantageous to employ nen when it becomes necessary to draw vertically from below ds : and on the contrary, it is more advantageous to employ >f a considerable weight, when it is required to lift up loads by 352 ANIMAL STBBNOTH: SCHULZE's EXPERIMENTS. [PABT II. means of a pulley about which a cord passes, that the workmen may draw in a vertical direction, from above downwards. To find the absolute strength of these men in a horizontal direction, Mr. Schulze proceeded thus : — Having fixed over an open pit a brass pulley, extremely well made, of 15 inches diameter, whose axis, made of well polished steel to diminish the friction, was ^ of an inch in diameter, he passed over this pulley a silk cord, worked with care, to give it both the neces- sary strength and flexibility. One of the ends of this cord carried a hook to hang a weight to it which hung vertically in the pit, whilst the other end was held by one of the 20 men, who in the first order of the following experiments made it pass above his shoulders; instead of which, in the second, he simply held it by his hands. Mr. Schulze had taken the precaution to construct this in such a manner that the pulley might be raised or lowered at pleasure, in order to keep the end of the cord held by the man always in a hori- zontal direction, according as the man wss tall or short, and exerted his strength in any given direction. He had made the necessary arrangements so as to be able to load successively the basin of a balance which was attached to the hook at the end of the cord which descended into the pit, whilst the man who held the other end of this cord employed all his strength with- out advancing or receding a single inch. The following tables give the weights placed in the basin when the workmen were obliged to give up, having no longer sufficient strength to sustain the pressure occasioned by the weight. To pro- ceed with certainty, Mr. Schulze increased the weight each time by five pounds, beginning from 60, and took the precaution to make this augmentation in equal intervals of time ; having always precisely a space of 10 seconds between them. The result of these observa- tions repeated several days in succession, is contained in the follow- ing tables. I. When the cord passed over the shoulders of the workmen : Order. lbs. Order. lbs. Order. Ibe. Order. lU. 1 95 6 100 11 95 16 95 2 105 7 115 12 100 17 100 3 110 8 105 13 110 18 90 4 100 9 95 U 90 19 100 5 105 10 90 15 110 20 100 CHAP. VII.] ANIMAL STRENGTH: MEN. II. When the cord was simply held before the man : 353 Order. Ibt. Order. lbs. Order. lbs. Order. lbs. 1 90 6 100 11 90 16 90 2 105 7 no 12 90 17 90 3 105 8 100 13 100 18 85 4 90 9 90 14 85 19 100 5 95 10 85 15 105 20 100 These two tables show that men have less power in drawing a cord before them than when they make it pass over their shoulders ; thej show, also, that the largest men have not always the greatest strength to hold, or to draw in a horizontal direction, by means of a cord. To obtain the absolute velocity of these twenty men, Mr. Schnlze proceeded as follows : — Having measured very exactly a distance of 12,000 Rhinland feet, in a plane nearly level, he caused these twenty men to march with a fair pace, but without running, and so as to continue during the period of four or five hours ; the following is the time employed in describing this space, with the velocity resisting for each of them. Order. 1 Time. Velocj Order. Time. Veloc Order. Time. Veloc 4018 4-94 1 8 4009 4-99 ! 15 3617 5-51 2 4112 4-85 9 40-20 4-96 1 16 41-28 4-82 3 3908 5-55 ' 10 40-51 4-90 ' 17 42-25 4-71 4 39-40 504 11 3617 5-51 18 40-19 4-98 5 3419 5-83 12 38-11 5-24 19 39-57 501 e 3511 5-68 13 3805 5-25 20 37-51 5-29 7 3807 5-25 14 3701 5-40 It is necessary to mention, with regard to these experiments^ that Mr. Schulze took care to place at certain distances persons in whom he could place confidence, in order to observe whether these men marched uniformly and sufiiciently quick without running. Having thus obtained not only the absolute force, but the absolute velocity also, of several men, he took the following method to deter- mine their relative force. He made use of a machine composed of two large cylinders of very hard marble, which turned round a vertical cylinder of wood, and moved by a horse, which described in his march a circle of 10 Rhinlaod feet. This machine appeared the most proper to make the subsequent experiments, which serve to determine the relative strength that the men had employed to move this machine, and which is used hereafter to determine which of Euler's two formulas ought to be preferred. To obtain this relative force, he took here the same pulley which 354 ANIMAL 8TRIN0TH : MBK. [PART II. served in the preceding experiments, by applying a cord to the vertical cylinder of wood, and attaching to the other end of this cord, which entered into an open pit, a sufficient weight to give suceseivelv to the machine different velocities. Having applied in this manner a weight of 215 lbs., the machine acquired a motion, which after being reduced to a uniform velocity, taking into account the acceleration of the weight, of the friction, and of the stiflfness of the cord, gave 2*41 feet velocity ; and having applied in the same manner a weight of 220 lbs., the resulting uniform motion gave a velocity of 2*47 feet. These two limits are mentioned because they serve as a comjmrison with what imme- diately follows : Mr. Schulze began these experiments with a weight of 100 lbs., and increased it by five every time from that number op to 400 lbs. He made this machine move by the first seven of his workmen, placing them in such a way that their direction remained almost always perpendicular to the arm on which was attached the cord which passed over their shoulders in an almost horizontal direction. Thus situated, they made 281 turns with this machine in two hourH, which gave for their relative velocity p = 2*45 feet per second. We have also the absolute force, or f, from these scTen men, by the above table, ^ 730 lbs. : and their absolute velocity or V = 5-30 feet. Therefore, by substituting these values in the first formola, we find the relative force F = 205 lbs., which agrees very well with what we have just found above. If instead of this first formula the second be taken, it gives F =: 153 lbs., which is far too little. By this it is evident, that the last of Ruler's two formulae is to be preferred in all respects. Mr. Schulze made a great number of com- binations, and almost alu'ays found the same effect. Dividing the 205 lbs. which we have just found by seven, the number of workmen, we get 29 lbs. for the relative force with 2'45 feet relative velocity for each man, which is rather more than the values commonly adopted in the computation of machinery. A number of other observations on different machines have given the same result; that is to say, we must value the mean human strength at 29 or 30 lbs., with a velocity of 2|feet per second. To obtain the ratio of the strei^gth of a horse to that of a mm, Mr. Schulze proceeded in a similar manner; but his results, in reference to that inquiry, are neither so correct nor so interesting. 8. In the first volume of my Mechania^ I stated the ayerage force of a man at rest to be 70 lbs., and his utmost walking velocity when unloaded, to be about six feet per second; and thence inferred that a man would produce the greatest momentum when drawing Sl^lbs* along a horizontal plane with a velocity of two feet per second. But that is not the most advantageous way of applying human strength. 9. Dr. Desagaliert asserts, that a man can raise of water or any other OHAP. VII.] ANIMAL STRENGTH : MEN. 355 weight aboat 550 lbs., or one hogshead (weight of the vessel included), ten feet high in a minute; this statement, though he says it will bold good for six hours, appears from his own facts to be too high, and 18 certainly such as could not be continued one day after another. Mr. Smeaton considers this work as the effect of haste or distress ; and reports, that six good English labourers will be required to raise 21141 cubic feet of sea water to the height of four feet in four hours: in this case, the men will raise a very little more than six cubic feet of freah water each to the height of 10 feet in a minute. Now, the hogshead containing about 8^ cubic feet, Smeaton s allowance of work proves less than that of Desaguliers in the ratio of 6 to 8^ or 3 to 4|. And as his good English labourers who can work at this rate are ettiinated by him to be equal to a double set of common men picked Qp at random, it seems proper to state that, with the probabilities of voluntary interruption, and other incidents, a man's work for several neoessive days ought not to be valued at more than half a hogshead laised 10 feet high in a minute. Smeaton likewise states, that two ordinary horses will do the work in three hours and twenty minutes, which amounts to little more than two hogsheads and a half raised 10 feet high in a minute. So that, if these statements be accurate, one horse will do the work of five men. Mr. Emerson affirms, that a man of ordinary strength, turning a roller by the handle, can act for a whole day against a resistance equal to 30 IbSs weight ; and if he works 10 hours a day, he will rsise a weight of 30 lbs. through 3^ feet in a second of time ; or, if the weight be greater, he will raise it to a proportionally less height. If two men work at a windlass, or roller, they can more easily draw up 70 lbs. than one man can 30 lbs., provided the elbow of one of the handles be at right angles to that of the other. Men used to bear loads, such as porters, will carry from 150 lbs. to 200 or 250 lbs., scGording to their strength. A man cannot well draw more than 70 Iba. or 80 lbs. horieontally : and he cannot thrust with a greater force acting horisontally at the height of his shoulders, than 27 or 80 lbs. But one of the most advantageous ways in which a man can exert bis force is to sit and pull towards him nearly horizontally, as in the action of rowing. M. CSoulorob communicated to the French National Institute the results of various experiments on the quantity of action which men csn afford by their daily work, according to the different manners in which they employ their strength. In the first place he examined the quantity of action which men can produce when, during a day, they mount a set of steps or stairs, either with or without a burden. He found that the quantity of action of a man who mounts without a burden, having only his own body to raise, is double that of a man loaded with a weight of 68 kilogrammes, or 150 lbs. avoirdupois, both eoDtinuing at work for a day. Hence it appears how much, with equal fatigue and time, the total or absolute effort may obtain different fdnes by varying the combinations of effort and velocity. But ihe word effect here denotes the total quantity of labour A A 2 356 ANIMAL STRENGTH: MBK. [PABT IT. employed to raise, not only tbe burden, but the man himself; and, as Coulomb observes, what is of the greatest ]m|K)rtance to consider is the useful effect,, that is to say, the total effect, deducting the value which represents the transference of the weight of the man's body. This total effect is the greatest possible when the man ascends without a burden ; but the uteful effect is then nothing : it is also nothing if the man be so much loaded as to be scarcely capable of movbg, and consequently there exists between these two limits a value of tbe load such that the useful effect is a maximum. M. Coulomb supposes that the loss of quantity of action is proportional to tbe load (an hypotliesis which experience confirms), whence he obtains an equa- tion which, treated according to the rules of maxima and minima, gives 53 kilogrammes (117 lbs. avoirdupois) for the weight with which the man ought to be loaded, in order to produce during one day, by ascending stairs, the greatest useful effect : the quantity of action which results from this determination has for its value 5(^ kilo- grammes ( 1 23| lbs. avoirdupois) raised through one kilometre, or nearly 1094 yards. But this method of working is attended with a loss of three-fourths of the total action of tlie man, and consequently costs four times as much as work, in which, after having mounted a set of steps without any burden, the man should suffer himself to fall by any means, so as to raise a weight nearly equal to that of his own body. From an examination of the work of men walking on a horizontal path, with or \%nthout a load, M. Coulomb concludes that the greatest quantity of action takes place when the men walk being loaded ; and is to that of men walking under a load of 58 kilogrammes (128 lbs. avoirdupois) nearly as 7 to 4. The weight which a man ought to carry in order to produce the greatest useful effect,, namely, that effect in which the quantity of action relative to the carrying his own weight is deducted from the total effect, is 50*4 kilogrammes, or 111*18 lbs. avoirdupois- There is a particular case which always obtains with respect to burdens carried in towns, viz., that in which the men, after having carried their load, return unloaded for a new burden. The weight they should carry in this case, to produce the greatest efiect, is 61*25 kilogrammes (135^ lbs. avoirdupois). The quantity of osefol action in this case, compared with that of a man who walks freely to^ without a load, is nearly as 1 to 5, or, in other words, he employs to pure loss ^ths of his power. By causing a man to mount a set of steps freely and without burden, his quantity of action is at leait double of what he affords in any other mediod of employing ^ strength. When men labour in cultivating the ground, the whole qoantitj afforded by one man during a day amounts to 100 kilogrammes ele- vated to 1 kilometre, that is, 220*6 lbs. raised 1094 yards. M. Coulomb, comparing this work with that of men employed to csrr; burdens up an ascent of steps, or at the pile-engine, finds i loM of about ^ th part only of the quantity of action, which may he neglected m researches of this kind. CHAP. VII.] ANIMAL STRBNGXH : IIBN. 357 In estimating mean results we should not determine from experi- ments of short duration, nor should we make any deductions from the exertions of men of more than ordinary strength. The mean results baye likewise a relation to climate. '' I have caused," says M. Cou- lomb, ^ extensive works to be executed by the troops at Martinico, where the thermometer (of Reaumur) is seldom lower than 20^ or 77^ of Fahrenheit). I have executed works of the same kind by \e troops in France, and I can affirm, that under the fourteenth degree of latitude, where men are almost always covered with perspir- ation, they are not capable of performing half the work they could perform in our climate." 10. Entirely according with these are the experiments of Regnier, by means of a dynamometer, the results of which not only established the superiority of civilized men over savages, but that of the English- man over the Frenchman. The following is reduced from one of Regnicr's tables of mean results. £ Strength. Savages, of Van Dicmen's Land New Holland . . . Timor Civilised men : French . . . English . . . With the With the hands. reins. Ibf. (m. lbs. OS. 30 6 51 8 14 8 58 7 10 2 69 2 22 1 71 4 23 8 11. A porter in London is accustomed tu carry a burden of 200 Ibe. at the rate of three miles an hour; and a couple of chairmen continue at the rate of four miles an hour, under a load of 300 lbs. Yet these exertions, Professor Leslie remarks, are greatly inferior to the subanltory labour performed by porters in Turkey, the Levant, and generally on the shores of the Mediterranean. At Constan- tinople, an Albanian porter will carry 800 or 900 lbs. on his back, stooping forward, and assisting his steps by a short staff. Such loads, however, are carried for very short intervals. At Marseilles it is affirmed that four porters carry the immense load of nearly two tons, by means of soft hods passing over their hcads^ and resting on their shoulders, with the ends of poles from which the goods are sus- pended. 12. With regard to the magnitude of the comparative efforts of man in different employments, the late Mr. Robertson Buchanan ascertained, that in working a pump, in turning a winch, in ringing a bell, and rowing a boat, die dynamic results are as the numbers 100, 167, 227, and 248. According to the interesting experiments described in M. Hachette's Traiie des Mctckinesy the dynamic unity being Uie weight of a cubic metre of water raisid to the height of one metre [that is, 2208 lbs. avoirdupois, or 4 hogsheads raised to the height of 3*281 feet, or 356 ANIMAL STRSlfeTH: lfEK» [PABT 1*3124 hogsheads to the height of 10 feet], we have the folio? measures, at a medium, of the daily actions of men. Dyn. I 1. A man marching 7| hours on a slope of 7 degrees with a load of from 15 to 18 lbs = 2. Marching in a mountainous country without load = 3. Carrier of wood up a ladder, his weight 123 Ibs^ his load llTlbs = 4. Carrier of peat, up steps, his own weight compnced, 112 to = 5. Man working at the cord of a pulley to raise the ram \ of a pile engine: three examples ) 6. A man drawing water from a well by means of a cord as 7. Man working at a capstan := 8. Man working at a capstan to raise water, mean of 24 observations ! = The unit of transport being the weight of a cubic metre of w carried a metre (or 2208 lbs. carried 3281 feet) upon a horizo road, we have for the daily action, Dyn. T 1. A man travelling without load on a flat road, his weight 1 54 lbs. his journey 31 ^ miles ^ 3 2. A soldier, carrying from 44 to 55 lbs., travelling 12| miles, 1800 to = 1 3. Ditto, a forced march of 25 miles = 2 4. A French porter, weight of the man not included, 792 to =« 5. Porter with wheel-barrow, weight of the man not included b: 1 6. Porters with a sledge ss 7. A man drawing a boat on a canal; 110310 lbs. coq- veyed 6 J miles * xr= 550 14. Mr. B. Bevan, an able engineer, has made experiments on application of human energy to the use of augurs, gimlets, scr drivers, &c. He has presented to the public the following list, ^)echnen ; premising that many ordinary operations are performe a short space of time, and may therefore be done by greater exei than if a longer time was necessary. Thus a person, for a a time, is able to use a tool or instrument called lbs. A drawing-knife, with a force of 100 An augur, with two hands 100 A screw-driver, one hand 84 A common bench vice handle 73 A chisel and awl, vertical pressure 72 A windlass, handle revolving 60 CHAP. VII.] ANIMAL STBBNOTH : MEN. 35^ lb«. Pincers and pHers, compression 60 A hand-plane, horizontallj 50 A hand or thamb-vice 45 A hand-saw 36 A stock-bit, revolving 16 Small screw-drivers, or twisting by the thumb and fore-finger only 14 15. M. Morisot informs us that the time employed by a French stone-mason's sawyer, to make a section of a square toise (40*89 square feet English) in different stones, is as below : viz. hours. Calcareous stone, equal grain, spec. grav. 2000 45 hard, spec. grav. 2300 62 Liais, ditto hard, fine grain, spec. grav. 2400 67 Pyrenean alabaster^ the softest of the marbles 56 Normandy granite 504 Granite from Vosges 700 Red and green porphyry 1177 The workmen ordinarily made 50 oscillations in a minute ; each stroke about 15| inches. 16. Hassenfratz assigns 13 kilogrammes as the mean effort of such a man ; but M. Navier, in his new edition of BdidoTy Architecture Hydrau- Uque^ regards this estimate as too high. If Hassenfratz were correct, the daily quantity of action of the sawyer would be equivalent to 376 kilogrammes elevated to a kilometre (or 818 lbs. raised g of a mile), a quail titv more than triple that of a man working at a winch. M. Navier gives, as a more correct measure of this labour for 1 2 hours, 188 kilogrammes raised a kilometre : half the former measure. But «U this is probably very vague. 17. Among quadrupeds, those which are employed to produce a mechanical effect are the dog, the ass, the mule, the ox, the camel, and the horse. Of these the horse is the only one, so far as we are aware, whose animal energy has been subjected to cautious experi- ments ; and, even with regard to this noble animal, opinions as to actoal results are very much afloat. The dynamic effort of the horse is, however, probably about 6 times that of. a strong and active labourer. Desaguliers states the proportion as 5 to 1, coinciding with the deductions of Smeaton. The French autliors usually regard sevep men as equivalent to one horse. As a fair mean between theae, I assumed in vol. i. of my Mechanics the proportion of 6 to 1, and stated the strength of a horse as equivalent to 420 lbs. at a dead pull. But the proportion must not be regarded as constant, but obviously varies much according to the breed and training of the animal, as well as according to the nature of the work about which he is employed. Thus the worst way* as De la Hire observed, of 860 ANIMAL STRBNOTH: H0R8B8 [PABT II. applying the strength of a horse is to mal«c him carry a weight up a steep hill ; while the organization of a man fits him very well for tiiat kind of lahour: hence three men, climhing up such a hill with a weight of 100 Ihs. each, will proceed faster than a horse with a load of 300 Ihs. 18. In the memoirs of the French Academy for 1703 are inserted the comparatiye ohservations of M. Amontons, on the velocity of men and of horses ; in which he states the velocity of a horse loaded with a man and walking to he rather more than 5| feet per second, or 3 1 miles per hour, and when going a moderate trot with the same weight to he ahout 8J feet per second, or ahout 6 miles per hour. These velocities, however, are somewhat less than what might have heen taken for the mean velocities. 19. But the best way of applying the strength of horses is to make them draw weights in carriages, &c. To this kind of labour, therefore, the inquiries of experimentalists should be directed. A horse put into harness, and making an effort to draw, bends himself forward, inclines his legs, and brings his breast nearer to the earth ; and this so much the more as the effort is the more considerable. So that, when a horse is employed in drawing, his effort will depend, in some measure, both upon his own weight and that which he carries on his back. Indeed it is highly useful to load the back of a drawing horse to a certain extent; though this, on a slight consideration, might be thought to augment unnecessarily the fatigue of the animal : but it must be recollected that the mass with which the horse is charged vertically is added in part to the effort which he makes in the direction if traction, and thus dispenses with the necessity of his inclining so much forward as he must otherwise do : and may, therefore, under this point of view, relieve the draught more than to compensate for the additional fatigue occasioned by the vertical pressure. Carmen, and waggoners in general, are well aware of this, and are commonly very careful to dispose of the load in such a manner that the shaf)^ shall throw a due proportion of the weight on the back of the shall horse. This is most efficaciously accomplished at Yarmouth, in Nor- folk, where a number of narrow streets connecting the market-place i^dth the quay, have led to the invention and use of the low, strong, narrow carts, thence denominated Yarmouth carts, drawn by one horse; and on which the loads are frequently shifted, especially when the vehicles pass over the bridge, in order to give the animals better foot-hold, and consequently a greater dynamic effort. • 20. The best disposition of the traces during the time a horse is drawing is perpendicular to the position of the collar upon his breast and shoulders : when the horse stands at ease, this position of the traces is rather inclined upwards from the direction of the road ; bat when he leans forward to draw the load, the traces should then be- come nearly parallel to the plane over which the carriage is to be drawn ; or, if he he employed in drawing a sledge, or any thing wiili • out wheels, the inclination of the traces to the road should (from the CHAP. Vn.'] ANIMAL 8TRKNGTH : HOftSIS. 361 table at page 52QO) be about 18|% when the friction is one-third of die presfiure. If tlie relation of the friction to the pressure he dif- ferent from this, the same tahle will exhibit the angle which the traces most make with the road. 21. When a horao is made to more in a circular path, as is often pnctised in mills and other machines moved by horses, it will bo necesMry to give the circles which the animal has to ^'alk round the greatest diameter that will comport with the local and other con- diuuns to which the motion roust be subjected. It is obvious, indeed, tLat, since a rectilinear motion is the most easy for the horse, the less the line in which he moves is curved, with the greater facility he will walk over it, and the less he need recline from a vertical |)osition : and besides this, with equal velocity the centrifugal force will be less b tbe greatest circle, which will proportionally diminish the friction of tbe cylindrical part of the trunnions, and the labour of moving the machbe. And, further, the greater the diameter of the horse- walk, tbe nearer the chord of the circle in which the horse draws is to coin- cidence with the tangent, which is the most advantageous position of tbe line of traction. On these accounts it is that, although a horse man draw in a circular walk of 18 feet diameter, yet in general it is advisable that the diameter of such a walk should not be less than 25 or SO feet ; and in many instances 40 feet would be preferable to either. ii. It has been stated by Desaguliers and some others, that a borse employed daily in drawing nearly horizontally can move, during cigbt hours in the day, about 5iOO lbs. at the rate of 2i miles per hoar, or 3| feet per second. If the weight be augmented to about 240 or 250 lbs., the horse cannot work more than six hours a day, and that with a less velocity. And, in both cases, if he carry some weight, he will draw better than if he carried none. M. Sauveur estimates the mean effort of a horse at 175 French, or 189 avoirdupois pounds, with a velocity of rather more than three feet per second. But all these are probably too high to be continued for eight hours, &T after day. In another place Desaguliers states the mean work of a horse as equivalent to the raising a hogshead full of water (or 550 lbs.) 50 feet high in a minute. But Mr. Smeaton, to whose authority much is due, asserts, from a number of experiment*, that the greatest effect is the raising 550 lbs. forty feet high in a minute. And, from wne experiments made by the Society for the Encouragement of Arts, under the direction of Mr. Samuel Moore, it was concluded, that a horse moving at the rate of three miles an hour can exert a force of 80 lbs. Unluckily, we are not sufficiently acquainted with tlw nature of the experiments and observations from which these MnctioDs were made to institute an accurate com]iarison of their '«*Qlt». Neither of them ought to express what a horse can draw "pon a carriage ; because in that case friction only is to l)e overcome («fter the load is once put into motion); so that a middling horse, well applied to a cart, will often draw nmcli more than 1000 lbs. The ^^^ estimate would be that which measures the weight that a 3G2 ANIMAL STBSKGTH: H0B8B8. [PABT n. horse wonld draw up out of a well ; the animal acting hj a horizontal line of traction turned into the yertical direction hj a simple pulley, or roller, whose friction should he reduced as modi as poesihle. 23. Mr. Tredgold, in his valuahle puhlication on Railroads, has directed his attention to the suhject of *'*'hor9e power" The follow- ing is his expression for the power of a horse, 250 ^ \l ) ; and \ V / for the day's work in Ihs. nused one mile ; d 1 -f « heing the hours which the horse works in a day, and the weight of 14*7 the carriage to that of the load as n : 1. He also gives -r for the a/ d greatest speed in miles per hour, when the horse is unloaded. These expressions must, at present, he regarded as tentative. The follow- ing is his tahlc of the comparison of the duration of a horse's daily labour and maximum of velocity, imloaded. Duration of labour. Max. Telocity unloaded Hours, in mile* per hour. 1 147 2 10-4 3 8-5 4 7-3 5 6-6 6 6-0 7 6-5 8 5-2 9 4-9 10 4-6 Taking the hours of labour at 6 per diem, the utmost that Mr- Tredgold would recommend, the maximum of useful effect he assigns at 125 lbs. moving at the rate of three miles per hour, and regardiiif the expense of carriage, in that cose, as unity ; then — Miles per hour. Proportional expense. Moving fbroe. 2 li or M25 166 lbs. 3 1 125 3J 1^7 or 1-0285 104 4 li or 1125 83 4j l| or 1-333 62j 5 l| or 1-8 41$ 5i 2 36J That is, the expense of conveying goods at 3 miles per hour bang 1 ; the expense of 4| miles per hour will be 1^ ; and so on, the eX' pense being doubled when the speed is 5^ miles per hour. 24. Thus, according to Mr. Tredgdd^ wc have for the day of CfiAK TU.] ANIMAL 8TB1N0TH *. HOBSES. 363 6 hours 2d60 lbs. raised one mile. And Mr. Bevan^ who has made many experiments on the force of traction to move canal boats on the Grand Junction Canal, found the force of traction 80 lbs., and the space travelled in a day 26 miles ; hence, it is only equivalent to 26 X 80 = 2080 lbs. raised one mile for the day's work ; the rate of travelling being 2*45 miles per hour ; and the result a little less than Mr. Tredgdd'Sy the difference probably arising from the devia- tion of the angle of the catenary formed by the rope from the horizon. 25. The following experimental data from Mr. Bevan also deserve attention. ** In the period from 1803 to 1809, I had the opportunity of as- certaining correctly the mean force exerted by good horses in draw- ing a plough ; having had the superintendence of the experiments on that bead at the various ploughing matches both at Woburn and Ash- ridge, nnder the patronage of the Duke of Bedford and the Earl of Bridgewater. I find among my memoranda the result of eight ploughing matches, at which there were seldom fewer than seven teams as competitors for the various prizes. lbs. The first result is from the mean force of each horse in six teams, of two horses each team, upon light sandy soil = 156 The second result is from seven teams of two horses each team, upon loamy ground, near Great Berkhampstcad... = 154 The third result is from six teams of four horses each team, with old Hertford^ire ploughs =: 127 The fourth result is from seven teams of four horses each team, upon strong stony land (improved ploughs) = 167 The fifth result is from seven teams of four horses each team, upon strong stony land (old Hertfordshire ploughs) := 103 The sixth result is from seven teams of two horses each team, upon light loam = 177 The seventh result is from five teams of two horses each, upon light, sandy land = 170 The eighth result is from seven teams of two horses each team, upon sandy land = 160 " The mean force exerted by each horse from fifty-two teams, or one hundred and forty-four horses, equals 163 pounds each horse, and although the speed was not particularly entered, it could not be less than the rate of two miles and a half an hour. *^ As these experiments were fairly made, and by horses of the common breed used by farmers, and upon ploughs of various coun- Uea, these numbers may be considered as a pretty accurate measure of the force actually exerted by horses at plough, and which they are able to do without injury for many weeks; but it should be remem- bered, that if these horses had been put out of their usual walking pace, the result would have been very different. The mean power of the draught-horse, deduced from the above-mentioned experi- ments, exceeds the calculated power from the highest formula of 364 canals: railboaos. [pari II. Mr. Leslie^ which is as follows : (15 — o)^ = lbs. aToirdnpois for the traction of a strong horse, and (12 — v)* ^= lbs. traction of the ordinary horse, v = velocity in miles per hour." TABLE l. — Shomtig the maximum quantity of labour a Hone of averctye strength is capable of performing^ cU different velocities^ on Canals^ Railways, and Tumptke-roads. {From Tredgold.) Useful eflfect of one horse workinir one Duration day, in tons drawn one mile." I Velocity In of the day'* work at the 1 miles per traction In hour. preceding lbs. On a lerel velocity. On a canal. railway. level turn- pIke-roML milc«. houn. lbs. tons. tons. tons. ^ u\ 83i 520 115 14 3 8 83i 243 92 12 3J 4 83i 153 82 10 4 83i 102 72 9 5 a/o 83i 52 57 7-2 6 2 83J 30 48 60 7 U 83i 19 41 51 8 n 83^ 12-8 36 4-5 9 ^?. 83i 90 32 40 10 H 83i 6-6 28-8 3-6 Where horse power is employed for the higher velocities, the ani- mals ought to be allowed to acquire the speed as gradually as pos- sible at the first starting. This simple expedient will save the pro- prietors of horses much more than they are aware of; and it deserves their attention to consider the best mode of feeding and training horses for performing the work with the least injury to their animiu powers. To compare the preceding table with practice at the higher velo- cities, it will be necessary to have the total mass moved, which is one-third more than the useful effect in this table. Now, the actual rate at which some of the quick coaches travel, is 10 miles an hour; the stages average about 9 miles ; and a coach with its load of lug- gage and passengers amounts to about 3 tons ; therefore the average day's work of 4 coach horses is 27 tons drawn one mile, or 6| tons drawn one mile by one horse. The table gives 3*6 tons, addea }- of 3*6 = 4*8 tons drawn one mile for the extreme quantity of labour for a horse at that speed, upon a good level road ; from which should be deducted the loss of effect in ascending hills, heavy roads, &C., which will make the actual labour performed by a coach-horse ave- rage about double the maximum given by the table. The conse- quences are well known. Accordinjr to Mr. Bevans observations, the horses on the Grand CftAP. VII.] TRACTION ON CANALS, RAILROADS. 365 Jonotion Canal draw 617 tons one mile, at the velocity of 2*45 miles per hour. According to Mr. TrtdgM^ if V he the maximum velocity of a horse, and id any other velocity, the immediate power of a horse is 250 r f I "- T7 ) ; and, when the weight of the vessel or carriage is to the weight of the load, as n: 1, we have V V / =: 1 H-n the effective power ; and d being the hours the horse works in one day, the day's work will be V VV in lbs. raised 1 mile, 1 +n and 250 f I ^ -1 J = the force of traction in lbs. But if the force 14*7 were immediately applied, the value of V would be —7-); and to find the value when the waggons alone are moved, we have 1 147 14-7 1 : / , : : --— , : ~" 7~:., — ; = V ; whence the day's work v/l + « s/ d y/d (1 + n) ' ^ , fl60dv / v^d(\-^n)\ .... . , 18 I 1 -—\-)-= ' ] I which IS a maximum when 1 + n \ 14-7 /' 06 = d. Consequently, when the velocity is given, we r* (1 + «) 06 bftve -77- r equal the duration of the day's work in hours: tr(l -f 11) ^ 8000 / 0*8 \ and — rr Tj- = the effective day's work; and 250 f I — -— j = 83^ lbs. But we may assume n to be always so near ^, as not to affect the result : and then, —- = rf, and = the day's work in 2 lbs. or very nearly - tons raised one mile. This, being combined with the Dumbers of the preceding table, gives the effect of a horse on canals, railroads, and turnpike-roads. It must, however, be here added, that although the deductions from Mr. Tredgdd^% valuable tables, as to the effects on canak^ are tolerably accurate up to rates of 4 or 5 miles per hour, yet, when boats are moved on canals at rates of from 9 to 12 or 14 miles per hour, the circumstances of the resistances undergo an essential ehaoge. The resistance, in fact, becomes so small, that passage- 366 TaACTION ON CANALS, RAILBOADS. [PART II. boats DOW trayel at these high velocities; and it is hence probable that railroads and canals will admit of a competition such as the sup- porters of railroads never anticipated. I shall here briefly detail some of the facts, as they have been given in a letter widely circulated by Mr. W. Grakame of Glasgow, in the Nautical Magaziney and other places. From the traffic by canal boats, which has been actually going on during the last two years and a half, on the Paisley canal, we learn this remarkable fact, that, while a speed of ten miles per hour has been maintained by the canal boats, the banks have sustained no in- jury whatever. The cause of injury, in truth, has been entirely sup- pressed by the velocity of the boat, which passes along the water without raising a ripple. About two years ago, measures were adopted for increasing the speed of the boats on the Paisley or Ardrossan canal. This canal is by no means favourable to such experiments, being both serpentine in its course, and narrow : it connects the town of Paisley with the city of Glasgow, and the village of Johnstone ; the distance being about twelve miles. The boats employed on this canal are 70 feet in length, and 5*6 broad, and carry, if necessary, upwards of 120 passengers. They are formed of light iron plates, and ribs covered with wood, and light oiled cloth, at a whole cost of about 1 25/. They perform stages oJT four miles in an interval of time varying from 22 to 25 minutes, including all stoppages, and the horses run three or four of these stages alternately every day. The passengers are under cover, or not, as they please, no difference being made in this parti- cular ; and the fare is one penny per mile in the first, and three farthings per mile in the second cabin. The horses drawing the canal boat are guided by a boy, who rides one of them ; and, in passing under bridges at night, a light is shown in the bow of the boat, by which he sees his way, and which light is closed when the bridge is passed. Intermediate passengers are also accommodated to distauces even as small as a mile ; and the faci- lity with which the boat stops, when reHeved from the drawing force, is such as avoids all danger whatever. The expense of conveying a load of eight tons at a rate of nine or ten miles per hour^ including all outlay, interest, and replacement of capital, is not more than i\d. per mile. It is also ascertained that one ton weight may be carried on a canal at nearly the same speed as on the railway, at about 1^ per mile, including an allowance for interest and replacement of capital. It is also believed, that if the breadth and curvature of the Paisley canal admitted boats of 90 feet length, instead of 70, they would carry more passengers by one-half without an additional expense, and a decrease of labour to the horses. The foregoing has been deduced from calculations founded oo the observation of facts relating to the wear and tear of boats and horses, and the absolute resistance which these boats meet with in passing through the water. On this subject it has been observed. CHAP. VU.] TRACTION ON CANALS, RAILROADS. 3^7 that, in addition to the common resistance of the water to the motion of the boat, a wave, or body of water, is also raised before it, varying in its height according to the velocity of the boat, and constantly presenting an obstacle to ber progress, providing that she only moves through the water at a certain slow rate. The height of this wave will then amount to nearly two feet, often overflowing the banks of the canal, and, from the obstruction it occasions, eventually obliging the boat to be stopped. Now, if, instead of stopping the boat when this wave is raised, her velocity be increased beyond what it had then been, she advances and passes over it, and leaves it to subside in her wtike, which it does, and the water becomes perfectly still. The same horses, drawing the boat at this increased speed, are found to perform their work better, the resistance to their progress having become less ; and the more the velocity of the boat is thus increased, the less resistance she meets with, merely having to cut the still water instead of the wave. It is a curious fact, that the wave produced by the approach of a slow canal boat is often observed at the distance of a mile, and upwards, along the canal, before the arrival there of the boat. But, in the case of the high wave being raised by the Paisley canal boat, it is customary to stop the boat, and after it has subsided to start again at a greater velocity. When the boat 18 to be stopped for any purpose, as her speed decreases the wave rises in proportion, and washes over the banks, until the motion of the boat becomes so small Iks to produce none. The discovery is doahtless a very important one, and, if turned to account, is likely to produce a material alteration in the rate of transport on canals. It was not known until these experiments were made, that if a boat, from a state of rest, was dragged along a canal, in proportion as her •peed increased to a certain limit, that the power required was greeier: bat that, if she were started at, and preserved a speed ex- ceeding the same limit, the power required would be less, and would decrease as her velocity increased, in fact, from a certain velocity there seems to be no limit to the rate at which a boat, as far as ani- mal power can be applied, may thus pass through the water ; and as the rate increases the power required decreases. On this principle it is that the boats on the Paisley canal, with ninety passengers in them^ are drawn by horses at a speed of ten miles an hour; while it would kill them to draw the same boat along the canal at six mile$ an hour. A boat might indeed travel fifteen or twenty miles an hour eancr than at six miles. The former of these velocities has already been attained by Mr. Grahamey along a distance of two miles, and is considered by him safer both for the boat and tlie canal. As a proof of what may be done by this method of carriage, Mr. Chrakame states that he has performed a voyage of fifty-six miles along two canals in six hours and thirty-eight minutes, which in- cluded the descent of ^ye^ and the ascent of eleven locks, the pas- sage of eighteen drawbridges where the tracking-lino was thrown off, and sixty common bridges, besides a tunnel half a mile long ; all of course prodacing some delay. The boat which performed this 368 TRACTION UN CANALS AND ROADS. [PART II. was sixty-nine feet long, and nine broad, drawn by two borses, and carried thirty-three passengers, with their luggage and attendants. These facts furnish great encouragement to canal companies, to improve the construction of their boats and the speed on their canals ; and thus, probably, in some situations, supersede the necessity of railroads. Mr. Macneill, the assistant engineer upon the Holyhead road, under Mr. Telford, in the course of his examination before a committee of the House of Commons, on steam-carriages, railroads, &c. gave the subjoined curious information. Well-made roads, formed of clean hard broken stone, placed on a solid foundation, are little affected by changes of atmosphere ; but weak roads, or such as are imperfectly formed i^dth gravel, flint, or round pebbles, without a bottoming of stone, pavement, or concrete, are much affected. On the generality of roads, the proportional injury from the wea- ther and traffic is nearly as follows. When travelled by fast coaches : from atmospheric changes 20 ; coach-wheels 20 ; horses' feet 60 = 1 00. When travelled by waggons : atmospheric changes 20 ; waggon- wheels 35'5 ; horses' feet 44*5 = 100, Has ascertained, from a num- ber of observations, that the wear of the iron tire of fast-going coach- wheels is, compared with that of the shoes of the horses which draw them, as 326*8 to 1000, or as 1 to 3-4ths nearly ; and infers that the comparative injury done by them to roads is nearly in the same pro- portion. In the case of slow-going carriages and horses the propor- tion is as 309 to 360, or as 1 to 1*16, or nearly 1 to 1|. The tire of the wheels of the fast-going coaches last from two to three months, according to the weather, the workmanship, and quality of iron; about 20 years ago, it did not lost seven days on an average. Coach- horse shoes remain in use about thirty doys; waggon-horse shoes about five weeks on an average. Where roads are weak, and yield under pressure, the injury caused by heavy wheels is far greater than on solid firm roads. It was found, in one instance, that the wear of haid stone, placed on a wet clay bottom, was four inches, while it was not more than half an inch when placed on a solid dry foundation. On the Highgate archway road, the annual wear is not more than half an inch in depth. To the same gentleman we owe the following useful table. TABLE II. — Tlie general Result of Experiments made with a Stage Coachy weighing^ exdtuive of seven passengers^ 18 ctct.^ on the same piece of road^ on different inclinations^ and at different rtUes of velocitgy furnish the following statement. Rate of Inclination. Rates of Travelling. Force required. I in 20 6 miles per hour 268 lbs. 1 in 26 6 213 1 in 30 6 165 CHAX». VII.] Hate of Inclination. 1 TRACTION ON ROADS. 369 Rates of Travelling. Force required. n 40 6 miles per hour 160 lbs. In 600 6 Ill 20 26 30 40 in 600 8 20 26 30 40 10 10 10 10 n600 10 296 219 196 166 120 318 225 200 172 128 B B 370 STBEKOTH OF MATERIALS. [PART II- CHAP. VIII. STRENGTH OP MATERIALS. Sect. I. EestdU of Experiments^ and Principles upon which tkjf should he practically applied. By most writers on the strength of materials, it has been customarj to start with the assumption of the three under-mentioned principles, and upon them to construct a theory by means of which they afte^ wards deduced from experiments on rectangular bars tlie strength of beams of the same material, but of various forms of transTerse sec- tion. Upon, however, testing these deductions by experiment, thcj are found in many cases to be entirely erroneous, and to such an extent as to render their practical application not only uncertain, bat frequently highly dangerous. Such being the result of experience, it becomes important to examine these principles more closely, in order to discover wherein the cause of this erroneous result consists. The principles which we allude to are (in the language of Tred- gold*) as follows: — " The first is, that the strength of a bar or rod to resist a given strain, when drawn in the direction of its length, is directly pro- portional to the area of its cross section ; tchile its Mastic pc^n^ remains perfect^ and the direction of the force coincides wiUi the axis. " The second is, that the extension of a bar or rod by a fore* acting in the direction of its length is directly proporUonal to tbe straining force, when the area of the section is the same ; whiU ^ strain does not exceed the elastic power. " The third is, that while the force is within the elastic power of^ material^ bodies resist extension and compression with equal forces. It should be remarked, that each of the foregoing propositions i« only asserted to hold good " while the strain does not exceed the elastic power of the material." By which is meant that tbe force applied shall not be sufficient to cause any permanent displacement of the particles, and that when the straining force is removed, tbe body will not have acquired any set, but will reassume its former sbspc * Practical Treatise on the strength of cast iron. Fourth edition, pw ISi STRSNOTH OF MATBKIALS. 371 ». Tredgold and most other writers upoD this subject d that no permanent displacement or set takes place ining force amounts to about one-third of tho ultimate be material, and have consequently assumed that these ght be applied without error to determine the strength sv of materia], of any form, and subject to any kind more careful experiments have, however, shown conclu- permanent displacement or set takes place with a much I than one-third of the ultimate strength, and that in able set is produced by a straining force very much less which the material will be likely to be exposed in cfa being the case, it is obvious that these principles (as »Te) are inapplicable to any practical case. As, how- ire been disposed to imagine that such a strict limitation icmtion is not necessary, and that they ])ractically hold ben the force applied eaxeeds the elastic power of the ill be desirable to show that not only is such a supposi- nroneous, but that the principles themselves, even with t, so far from being demonstrable, are, on the contrary, loubtful. ■ertion, that the strength of a rod or bar subjected to n is directly proportional to the area of its cross section, irat sight it may appear obvious enough, nevertheless is practice to be strictly correct ; for, if the material is of it found that the strength of the exterior hard crust is bat of the interior substance, and therefore that the bicb the perimeter of the section bears to its area is a requiring to be taken into consideration ; not to men- urangement of the particles during the cooling of the ipon which arrangement the strength of the substance so lends, is greatly influenced by the external form of its i on the other hand, in the case of a fibrous material, aa da or timber, in consequence of the fibres not laying illel to each other, in cutting the bar to the required I may of the exterior fibres will be cut transversely, and > rfbre be capable of affording so great a proportionate li m similar fibres within the more central portion of the jj ieh it follows, that the tensile strength of a bar of any i directly proportional to its sectional area, although tho I » trifling as not to require attention in most of the cases ij i tfaemselyes in practice. { I law, that the extension of a bar or rod exposed to a \^ m directly proportional to that strain 90 long cu ike area \ remains unaltered^ is from the very circumstances of \ ■ible, since it is not capable for a bar to become elon- biTing its transverse section proportionately diminished, we cannot conceive the bar in two successive degrees I l^Ting in both the same sectional area. And although B B 2 372 STRENGTH OF MATBBIALS. [PABT IL the correct law subsisting between the elongation and the force pro- ducing it, has not yet been satisfactorily determined, there is eierr reason to believe that they are not directly proponionil to em other. With regard to the third law, which may be more definitely ititfld as follows, viz. : — that within the elastic power of the materaJ,bo&i require as great a force to alter their dimensions to any giTen exttnt by compression as by extension ; we may remark, that an? propeilf depending upon the straining force being within the elastie power tf the material is incapable of any practical application, and tbeRfim that this law, as limited above, even assuming it to be correet, doM not require to be included in any practical inquiry into the kvt which regulate the strength of materials. And without soch liniti^ tion the law is at utter variance with the results of experiment, net no fact is now better ascertained than that cast iron requires a dmIi greater force to compress it to any given extent, than to extend it to the same extent. Such then being the result of our investigation into the socoiteeB of these principles with practical experience, we cao no longer It surprised that the deductions drawn from them by Tredgc^ mI others, should be equally at variance with the same experieaefc Indeed the errors into which they have fallen appear to hate beet caused by reasoning upon hypothetical principles — attempthig to foia a complete theory of the strength of materials upon data neitlMr sufficient in quantity nor certainty. And even now, although una has been done, since the time when Tredgold wrote, by HodgkiMi Fairbaim, Thomas Cubitt, Barlow, and some others, this inraffieienef has not yet been supplied, and we are still in ignorance upon OMJ points, without a knowledge of which it would be impossible to fi»« any complete theory of the strength of materials aafiicient in Hi extent for all practical purposes, and of whose accuracy we cooM he so far sure as to allow of its application without fear of erroseov results. In order to show in what points the deficiency in oor knowle^ upon this subject consists, and to explain the general prindplei vp** which the strength of materials actually depends, we will hiiiif investigate the subject If A BCD, fig. 230, be a rectangular beam, subjected to fcj pressures Pj, P.^, and P.,, the two former of which arc eqoal to eidi other, and to half P,, and if Pi and P, be applied at the two eAUHJi ties of the beam, and P.< at its center in an opposite direetioBi ■• beam will be deflected, the upper surface A C becoming cooviX» •* the lower surface BD concave. Then, if the material compoiiogvt beam be supposed capable both of extension and com prcMi o^ w* fibres on the convex side will be extended, and those on the con** side compressed ; there will, however, be a certain curved aupeifc* £ F within the beam (called the neutral superficies), the fibres tfti^ in which will not be either extended or compressed. Now, if tangents EG, FU, to this curved neutral miperficiei it ^ CHAP. VIII.] STRENGTH OP MATEBIAL8. 373 extremities of the beam be drawn, and from them perpendiculars K L, M N, pasnng through the point I, in which the neutral superficies inter- lects the plane of rupture OQ; then will the triangle NIL be equal to the sum of the extensions of the whole length of all the extended fibres; tnd the similar triangle KIM will be equal to the sum of the com- pressions of the whole length of all the compressed fibres; and if any •operfides be taken between AC or BD and the neutral superficies, the portions of such superficies, as t^ i,, yi^s? contained between KL, and II N, will be equal to the extension or compression of the whole length of the fibres situate in that superficies : from whence it follows that the amount of extension or compression of any fibre is directly propor- tional to its distance from I, the neutral axis. If now any number of points be taken in the depth of the beam, as ij, f^ 1^ yj, ^29 y»* ^^' (%• 231), and lines be drawn through them perpendicular to O Q, each equal in length to the force required to extend or compress the fibres to the extent to which such fibres are actually extended or compressed, and which extent will be propor- tiona] to their distances from the neutral axis, and a curved superficies ^^i^iCtdid^e^l^bj^a^ be drawn through the ends of all these lines, then will the solid contained between the plane of rupture O Oj Q^ Q and the curved superficies, be equal to the resistance of all the extended and compressed fibres, or to the transverse strength of the beam. Now, if £ equals the solidity of ai^^lil^^^^a) ^^ ^ equals the soKdifcy of ^iCid^^d^c^I^ also ^and A equal the perpendicular distances of their center of gravity from the neutral superficies, and ^L the dis- tance of the points where the pressures P^ and F^ are applied, from the neutral axis ; we have from the principle of the equality of moments, iLP^ = E^ + CA; or, ainoe El must be equal to C^, and P^ equals \ P,, we have 5l'P,= E» (I.) ■ad this formula will apply» whatever form the transverse section of the beam may be, whether rectangular as fig, 281, cylindrical as fig. 232, or X shaped as fig. 233. It is further evident £at, when the beam is upon the point of breaking, the upper fibres at O must be strained to the utmost extent Id whkh they can be, before yielding, and that as this extent does not depend upon the form of the beam, but only upon the elastic properties of tne material, for the satee material the length a^ O will renuitn constant in all cases. From this it follows that the lines ■i, f^ •„ 7„ y^ 73, &c. (fig. 231), which are the ordinates of the enire a^hA^e^di^ will also be constant, but that their distances from E, or the Miacisse of the curve will vary directly as the distance O I of the neutral axis from the upper surface of the beam. In rectangular beams, or others, whose transverse sections are ■mikr figures, the distance O I, and therefore the area of the curve 374 STRBNOTH OP MATERIALS. [PART II. O a^ b^ Ip will yary as the depth of the beam, and the solidity of the figure Oa^b^ IjI^^^.a^O will vary as the depth and the breadth, or Eo: bd; also, since the forms of the beams are similar^ the distance i of the center of gravity of the solid E from the neutral superficies, mnst vary as the depth of the beam ; and therefore, in rectangular beams, the ultimate strength of the beam varies directly as the square of the depth multiplied by the breadth, and inversely as the length ; or, putting S (= Pj H- Pj rs P.^) for the breaking weight applied in the center,^ we have Sac^* (II.) or, when the transverse sections are similar figures^ </^ being any similar dimensions in both, d^ Sa^ (III.) From the preceding investigation it is evident that we cannot deduce any general formulse by which to determine the transverse strength of a beam of any given form of section, without knowing the position of the neutral axis, the limit to which the fibres may be extended before yielding, and the law which subsists between the extension and compression of the material and the force required to produce such extension and compression ; upon none of which sub- jects have we any certain knowledge. Sect. IL Stren^ ofMateriaU to resist tensile and crushing strains* Strength of Columns. The first principle alluded to in the foregoing section, via., that the strength of a bar or rod to resist a tensile strain in the direction of its axis, IS proportional to the area of its cross section, is not, for the reasons there mentioned, strictly correct. And although perhaps for ordinary practical purposes, or where the sections to be compared aie large, no error of importance wonld result from the adoption of this principle, it is very necessary, in experiments whose object is tbe determination of the actual strength of any given materia], that doe regard should be had to all circumstances, (however trivial they msj seem,) which may in any way affect the accuracy of the results. The only late experiments upon the tensile strength of cast iron, now before the public, are those of Mr. Eaton Hodgkinson, first published in a paper communicated to the British Association, and subsequently in his '^ Experimental Researches." The mean tensile strength of a square inch of cast iron, as deduced from these experi- ments, is only 10,560 lbs., or 7 tons 7*85 cwt.; the lowest heing 12,993 lbs., and the Iiighest 21,907 lbs. In a series of experiments, OHAP. VIII.] C0HE8IVB 8TRINOTH OF CAST IRON. 375 howeTer, which have been tried by Mr. Thomas Oubitt*, during a period of several years, we find the strength of cast iron to be con- siderably higher than Mr. Hodgkinson's estimate: the average of twenty-five specimens, of which eight were unsound, gave 27,140 lbs. for the tensile strength per square inch section ; and, exchiding the unsound pieces, the average of the others gave 27,773 lbs., or 12 tons 8 cwt. The highest broke with 32,997 lbs. per square inch, and the lowest^ even of the unsound pieces, broke with 21,471 lbs., or only a little under Mr. Hodgkinson's best speeimen. The anomaly between these results is very remarkable, since in both cases the experiments were conducted with the greatest care ; we think, however, that Mr. Cubitt's results are entitled to the pre- ference, for this reason, that while any cause might easily operate to lessen ihe strength of the iron, we cannot conceive it possible that any want of care in the experiments or bad arrangement of the apparatus could cause the iron to bear more than it otherwise would have done, providing that the means of measuring the force exerted in producing fracture were correct ; and these, we do not hesitate to aay, were not open to any exceptions. It is certainly difficult to account for so wide a difference in the results of these experiments, but we cannot but think that the form chosen by Mr. Hodgkinson, for the transverse section of the bars upon which he experimented, is open to objection. The form adopted by him is shown in fig. 234, being that of a cross ; and the reason which induced him to choose this form, was " to obviate the objection made by Mr. Tredgold (Art. 79 and 80,) and others, against the conclusions of former experimenters." The objection of Mr. Tredgold here alluded to was, that if the strain did not pass through the center of gravity of the section in the place where the rod broke, the different parts of that section would be exposed to a different strain, and therefore,' that the results obtained would be lower than the truth. We quite agree with Mr. Tredffold upon the importance of this objection, but we cannot bat consider, that the form chosen by Mr. Hodgkinson is more open to this objection than the square or cylindrical form would have been. For, whatever might be the form of section of the rod, the same means of fixing the ends of it might be used, and since it depends entirely upon the accuracy with which that is done whether the axis of the rod coincides with the straight line between the points of attachment, (which is the direction of the straining force), we may suppose that the lateral distance between these two lines at the breaking section would be the same, whatever was the form of that section; such * I am much indebted to Mr. Thomas Cubitt, for the liberal manner in wliich he has placed at my disposal the results of his very careful and ela- bormte experiments upon this subject. Ui« experiments on the set, deflection, and ultimate strength of cast iron girders, extend over a period of many years, and p o i Bs eis the immense practical advantage of having been made upon large girders. These experiments, amounting in number to upwards of 2000, have been taken under the direction of Mr. Dines; and, having witne98ed some of them myself, I can bear testimony to the care which has been taken to ensure aecurmte results, and to remove all imaginable causes of error.— H. L. 370 C0HE8IVB 8TRBN0TH OP MATERIALS. [PART II. being the case then, we hold that the strength of the cnicifonn sec- tion would be more affected than the square, since in the former the metal is situated further from what would then be the neutral axis of the section, and exposed therefore to greater differences of strain. This form is also open to the practical objection that it is very diffi- cult to obtain a perfectly sound casting from it ; it asually happens that the iron is not perfect at the point of intersection, (a, fig. 234,) and also to our remarks at p. 371, that the proportion of the peri- phery to the area may influence the strength. Mr. Cubitt has also tried a few experiments upon wrought iron, which gave for the average tensile strength per square inch, 58,952 lbs., or 26 tons 6*3 cwt. The metals differ more widely from each other, in their elastic force and cohesive strength, than the several species of wood or vege- table fibres. Thus, the cohesion of fine steel is about 135,000 lbs. for the square inch, while that of cast lead amounts only to about the hundred and thirtieth part, or 1800 lbs. According to the experiments of Mr. George Rennie, in 1817, the cohesive power of a rod an inch square of different metals, in pounds avoirdupois, with the corresponding length of the modulus of elasticity in feet, is as follows : — Pounds. Feet. Cast steel 134,256 ... 39,455 Swedish malleable iron ... 72,064 ... 19,740 English ditto 55,872 ... 16,938 Cast iron 19,096 ... 6,110 Cast copper 19,072 ... 5,003 Yellow brass 17,958 ... 5,180 Cast tin 4,736 ... 1,496 •Cast lead 1,824 ... 348 It thus appears, as Professor Leslie remarks, that a vertical rod o£ lead 348 feet long would be rent asunder by its own weight. Ther* best steel has nearly twice the strength of English soft iron, and this again is about three times stronger than cast iron. Copper and brass have almost the same cohesion as cast iron. This tenacity is some— times considerably augmented by hammering or wire-drawing, that= of copper being thus nearly doubled, and that of lead, according t o^^ Eytelwein, more than quadrupled. The consolidation is produced ^ chiefly at the surface, and hence a slight notch with a file will materially weaken a hard metallic rod. Professor Leslie has given the following tabular view of the lute cohesion of the principal kinds of timber, or the load which^ would rend a prism of an inch square of each ; and the altitude of tb^? prism which would be severed by the action of its own weight. Pounds. Feet. Teak 12,915 ... 36,049 Oak 11,880 ... d2,900 Sycamore 9,630 ... 35,800 Beech 12,225 ... 38,940 CHAP. VIII.] STRENGTH OF CORDAGE. 377 Pounds. Feet. Ash 14,130 ... 42,080 Elm 9,720 ... 39,050 Memelfir 9,540 ... 40,500 Christiana deal 12,34G ... 55,500 Larch 12.240 ... 42,160 The following is the result of Professor Barlow's experiments upon the cohesive strength of various kinds of timber. Pounds per square inch. Fir 12,203 Ash 17,077 Beech 11,467 Oak 10,389 Teak 15.090 Box 19,891 Pear 9,822 Mahogany 8,041 The cohesion of hempen fibres is, for every square inch of their transverse section, 6400 Ihs. The best mode of estimating the strength of a rope of hemp is to multiply by 200 the square of its number of inches in girth, and the product will express in pounds the practical strain it may be safely loaded with ; for cables, multiply by 120 instead of 200. The ultimate strain is probablv double this, as will appear from the account following of Du Hamei s experiments. If yams of 180 yards long be worked up into a rope of only 120 yards, it will lose one-fourUi of its strength, the exterior fibres alone resisting the greatest part of the strain. The register cordage of the late Captain Huddart exerts nearly the whole force of the strands, since they suffer a contraction of only the eighth part in the process of combining. For the utmost strength that a rope will bear before it breaks, a good estimate will be found by taking one-fi/lh of the square of the girth of the rope, to express the tons it will carrv. This is about double the role for practice which we have given above ; and is, even for an ulterior measure, too great for tarred cordage, which is always weaker than white. The following experiments were made by Mons. Du Hamel, at Rochfort, on cordage of three inches (French) circumference, made of the best Riga hemp, August 8th, 1741. White. Tarred. Broke with a strain of 4500 pounds «.. 3400 pounds. 4000 „ ... 3300 „ 4800 „ ... 3258 „ August 25th, 1743. 4600 „ ... 3500 „ 5000 „ ... 3400 „ 5000 „ ... 3400 „ 378 snunroTH op oobdaoi. [part ii. September 23, 1746. White. Tmrred. Broke with a itruo of 3880 pounds ... 3000 pounds. 4000 „ ... 2700 „ 4200 „ ... 2800 „ A parcel of wbite and tarred cordage was taken out of a quantity wLich had been made February 12, 1746. It was laid up in the Magazine, and comparisons were made from time to time, as follows : — Wbite. Tarred. Difference. 1746, April 1 4th, 2645 pounds 2312 pounds 333 pounds. 1747, Mav 18th, 2762 „ 2155 „ 607 1747, October 21st, 2710 „ 2050 „ 660 „ 1748, June 19th, 2575 „ 1752 „ 823 „ 1748, October 2nd, 2425 „ 1837 „ 588 „ 1749, Sept. 25th, 2917 „ 1865 „ 1052 „ M. Du Hamel says, that it is dcnnded by experience, that white cordage in continued service is one-third more durable than tarred ; secondly, it retains its force much longer while kept in store ; thirdly, it resists the ordinary injuries of the weather one- fourth longer. These obsenrations deserre the attention of pmctical men. Mr. B, Becan has favoured the author with a tabular view of his results with regard to the modulus of cohesion^ or the length in feet of any prismatic substance required to break its cohesion, or tear* it asunder. Bet>an% Remits, Feet. Tanned cow's skin 10,250 Tanned calf skin 5,050 Tanned horse skin 7,000 Tanned cordovan 3,720 Tanned sheep skin 5,600 Un tanned horse skin 8,900 Old harness of thirty years 5,000 Hempen twine 75,000 Catgut, some years old 23,000 Garden matting 27,000 Writing-paper, foolscap 8,000 Brown wrapping-paper, thin 6,700 Bent grass, (holcus) 79,000 Whalebone 14,000 Bricks, (Fenny Stratford) 970 Bricks, (Leighton) 144 Ice 300 Leicestershire slate 7,300 CHAP. VIII.] 8TRBN0TH OF COLUMNS. 879 STRENGTH OF COLUMNS. Upon the subject of the strength of columns, we are entirely indebted to Mr. Hodgkinson, who, after a very careful and accurate series of experiments, has deduced formulee of immense practical value, and has clearly shown the principles upon which the strength of materials, submit to a compressing force, depends. The results of his experiments are briefly as follows : — That when weights are applied to prisms, or cylinders, the shorter ones usually bore more than the longer ones, and that the strength may be con- sidered to vary as the area. That, when the diameter or least lateral dimension is less than the height of the prism, fracture takes place, by the separation of either a pyramid, cone, or wedee, (depending upon the form of the prism,) the angle of whose sides is constant for the same material, and about 34° for cast iron. And further, that the mode of fracture and the ultimate strength of a prism varies but slightly with a variation in its height, so long as it is not less than its diameter or least lateral dimension, or greater than four or ^ye times the same dimension; when higher, it begins to bend, and its strength decreases. Mr. Hodgkinson finds that the strength of a column is very much influenced by the manner in which the ends arc fixed ; when both ends are rounded, so that the column may turn upon them as a center, its strength is only one-third of that of another column of precisely similar dimensions, but whose ends are flat and incapable of motion ; and that, if one end is flat and immovable while the other is rounded, the strength is two-thirds of what it would have been had both ends been flat. The following table exhibits the results of his experiments on solid and hollow cylindrical columns of cast iron, in which S is the break- ing weight in tons ; D the external diameter, d the internal diameter, both in inches ; and / the length or height of the column in feet. Kind of Cdumn. With both ends round,! With both endii flat. when the height of the'when the height of the column is not less than'oolumn is not less than 15 times iu diameter. 30 times its diameter. Solid cylindrical cast ironl columns j Hollow ditto ditto S = 14-9 8 = 13 p3^ /1-7 8 = 44-16 8 = 44-3 /1-7 When the height of the column is less than that mentioned in this table, namely, fifteen times its diameter for columns with both ends rounded, and thirty times the same with both ends flat, the strength of the column becomes modified in consequence of its being then partially crushed as well as bent. If C equal the force which would be required to erush the column without flexure, 8 the strength as 380 8TRBN0TH OF COLUMNS. [PART II. calculated by the above formulse for long columns, and S^ the actual strength, we have '■-JTT' <>^-' The value of c, or the weight required to crush a square inch of cast iron, as deduced from his experiments, is 1 07,750 lbs., or 48 tons 2 cwt. ; therefore, the crushing force is to the tensile, according to Mr. Hodgkinson, as G'507 is to 1 ; or, taking Mr. Hodgkinson's value for the crushing strength, and Mr. Cubitt's for the tensile, the ratio becomes as 3*88 is to 1. The strength of a column of cast iron of given dimensions being 1000, the strength of a column of wrought iron of the same dimen- sions would be 1745, of cast steel 2518, of Dantzic oak 108*8, and of red deal 78-5. The weights required to crush cubes of the quarter of an inch of certain metals, according to the experiments of Mr. Rennie, arc as follows : — Pounds. Iron cast vertically 11,140 Iron cast horizontally 10,110 Cast copper 7,318 Cast tin 966 Cast lead 483 Cubes of an inch are crushed by the weights annexed : — Pounds. Elm 1,284 White deal 1,928 English oak 3,860 Craigleith freestone 8,688 Cubes of an inch and a half, and consequently presenting a section of two and a quarter times greater than the former, might be expected to resist compression in that ratio. They are crushed, however, with loads considerably less. Pounds. Red brick 1,817 Yellow baked brick 2,254 Fire brick 3,864 Craigleith stone, direction of the strata... 15,560 Ditto across the strata 12,346 White statuary marble 13,632 White-veined Italian marble 21 ,783 Purbeck limestone 20,610 Cornish granite ..« 14,302 Peterhead granite 18,636 Aberdeen blue granite 24,536 These facts show the comparative firmness of different materials ; but it is to be regretted that such results are not of much praetical value, since they are confined to a very narrow scale, and applicable CHAP. VIII.] MODULUS OP ELASTICITY. 381 only to cubical blocks. While the breadth remains the same, the resistance appears to depend on some unascertained ratio of the altitude of the column. Sect. III. EUutieiiy and ElongcUion of Bodies suhfected to a crushing or tensile strain. The modtdtis of the elasticity of any substance is a column of the same substance, capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length. The modulus of elasticity is the measure of the elastic force of any substance. A practical notion of the moduius of elasticity may be readily obtained. Let i be the quantity a bar of wood, iron, or other sub- stance, an inch square and a foot in length would be extended or diminished by the force f; and let / be any other length of a bar of equal base and like substance ; then 1 : / : : I : ^, or /f = ^, where A equals the extension or diminution in the length /. The modulus of elasticity is found by this analogy : as the dimi- nution of the length of any substance is to its length, so is the force that produced that diminution to the modulus of elasticity. Or, denoting the weight of the modulus in pounds for a base of an inch square by m, it is I :/: : 1 : w = '^. E And if v be the weight of a bar of the substance one inch square and one foot in length ; then, if M be the height of the modulus of elasticity in feet, we have M=-^ (V.) The weight of the modulus of the elasticity of a column being m, a weight bending it in any manner f the distance of the line of its application from any point of the axis D, and the depth of the column, dy the radius of curvature will be . The distance of the point of greatest curvature of a prismatic beam, from the line of direction of the force, is twice the versed sine of that arc of the circle of greatest curvature, of which the extremity is parallel to that of the beam. When the force is longitudinal, and the curvature inconsiderable, the form coincides with the harmonic curve, the curvature being pro- portional to the distance from the axis ; and the distance of the point 382 MODULUS OF ELASTICITY. [PABT II. of indifference from the axis becomes the secant of an arc propor- tional to the distance from the middle of the column. If a beam is naturally of the form which a prismatic beam would acquire, if it were slightly bent by a longitudinal force, calling its depth dy its length /, the circumference of a circle of which the diameter is unity c, the weight of the modulus of elasticity m, the natural deviation from the rectilinear form A, and a force applied at the extremity of the axis /y the total deviation firom the rectilinear form will be ^^VA^»_ It appears from this formula, that when the other quantities remain unaltered, t! varies in proportion to A, and if A = O, the beam cannot be retained in a state of inflection, while the denominator of the fraction remains a finite quantity; but when d^trm = 12 V f^ A^ becomes infinite, whatever may be the magnitude of A, and the force virill overpower the beam, or will at least cause it to bend so much as to derange the operation of the forces concerned. In this -^ j . -— , -8225 — w, which is the force capable of holding the beam in equilibrium in any inconsiderable degree of curvature. Hence, the modulus being known for any substance, we may determine at once the weight which a given bar nearly straight Is capable of supporting. For instance, in fir wood, supposing its height 10,000,000 feet, a bar an inch square and ten feet long may begin to bend with the weight of a bar of the same thickness, equal in length to -8225 x -— :: tt^ x 10,000,000 feet, or 671 feet; 120 X 120 that is, with a weight of about 120 lbs. ; neglecting the effect of the weight of the bar itself. In the same manner the strength of a bar of any other substance may be determined, either from direct experi- ments on its fiexure, or from the sounds that it produces. If ♦n /* / / = — , 'ri = -8225 «, and - = v' (8225 «) = 907 ^ n ; whence, n a a if we know the force required to crush a bar or column, we may calculate what must be the proportion of its length to its depth, in order that it may begin to bend rather than be crushed. When a longitudinal force is applied to the extremities of a straight prismatic beam, at the distance D from the axis, the deflection of the middle of the beam will be "•{-•-(V^-^)-^} ^"""-^ If a column, subjected to a longitudinal force, be cut out of a plank or slab of equable depth, in order that the extension and com- pression of the suifaces may be initially every where equal, its outline must be a circular arc. CHAP. VIII.] MODULUS OP ELASTICITY. 383 If a column be cut out of a plank of equable breadth, and the outline limiting its depth be composed of two triangles, joined at their bases, the tension of the surfaces produced by a longitudinal force will be ererj where equal, when the radius of curvature at the middle becomes eaual to half the length of the column ; and in this case the curve will be a cycloid. When the curvature at the middle differs from that of the cycloid, the figure of the column becomes of more difficult investigation. It may, however, be delineated mechanically, making both the depth of the column and its radius of curvature proportional always to ^/a. If the breadth of the column vary in the same proportion as the depth, they must both be every where as the culw root of a, the ordinate. ( Young 9 Nat, Phil, vol. ii.) The modulus of elasticity has not yet been ascertained in reference to so many subjects as could be wished. Professor Leslie exhibits several, however, as below. That of white marble is 2,150,000 feet, or a weight of 2,620,000 pounds avoirdupois on the square inch ; while that of Portland stone is only 1,570,000 feet, corresponding on the square inch to the weight of 1,530,000 pounds. White marble and Portland stone are found to have, for every square inch of section, a cohesive power of 1811 lbs. and 857 lbs. ; wherefore, suspended columns of these stones, of the altitude of 1542 and 945 feet, or only the ld94th and 1789th part of their respective measure of elasticity, would be torn asunder by their own weight. Of the principal kinds of timber employed in building and car- pentry, the annexed table will exhibit their respective Modulus of Elasticity, and the portion of it which limits their cohesion, or which lengthwise would tear them asunder. Teak 6,040,000 168th Oak 4,150,000 144th Sycamore 3,860,000 108tli Beech 4,180,000 107th Ash 4,617,000 109th Elm 5,680,000 146th Memelfir 8,292,000 205th Christiana deal .... 8,118,000 146th Larch 5,096,000 121th The following, also, exhibits Mr. Sevan s results as to the modulus of elasticity. Feet. Platinum 2,390,000 Gold (pure) 1,390,000 Steel 9,300,000 Bar iron 9,000,000 Ditto 8,450,000 Yellow pine 9,150,000 Ditto 11,840,000 384 MODULUS OF ELASTICITY. [PART II. Feet. Finland deal 6,000,000 Mahogany 7,600.000 Rose wood 3,600,000 Oak, dry 6,100,000 Fir bottom, 25 years old 7,400,000 Petersburg deal 6,000,000 Lance wood 5,100,000 Willow 6,200,000 Oak 4,360,000 Satin wood 2,290,000 Lincolnshire bog oak 1,71 0,000 Lignum Vitae 1,860,000 Teak wood 4,780,000 Yew 2,220,000 Whalebone 1,000,000 Cane 1,400,000 Glass tube 4,440,000 Ice 6,000,000 Limestone. „ Dinton, Buck 2,400,000 „ Kctton 1,600,000 „ Jettemoe 636,000 Ryegate 621,000 Yorkshire paving 1 ,320,000 Cork . 3,300 Slate, Leicestershire 7,800,000 The following is the weight of the modidus of elasticity of various substances employed very generally in construction, according to Mr. Tredgold. Pounds. Ash 1,640,000 Beech 1,346,000 Brass, cast 8,930,000 Elm 1,340,000 Fir, red or yellow 2,016,000 Fir, white 1,830,000 Iron, cast 18,400,000 Iron, malleable 24,920,000 Larch 10,740,000 Lead, cast 720,000 Mahogany 1,696,000 Oak, good English 1,700,000 Pine, yellow American 1,600,000 Steel 29,000,000 Tin, cast 4,608,000 Steel, cast 13,680,000 CHAP. Vni.] TaAN8V£B8B 8TBBN0TH OP CAST IBON. 385 Sbct. IV. On the Strength of Materials svhjected to a tramverse strain. From the obseirations which we have made in the first section, it appears that we are not possessed of all the data which are required for determining generally the strength of a cast-iron heam of any given form of section ; and consequently, that although rules have been given for that purpose, their results are in many cases, (as might be expected,) at utter variance with the actual strengths as obtained by experiment. Such, then, being the state of our knowledge upon the transverse strength of materials, we have abstained from giving any rules deduced alone from theory^ being well assured that to have an erroneims rule is much worse than to be without ani/ rule at all. Nevertheless, the strengths of a few forms having been determined by actual experiment, we are enabled to calculate with certainty, by means of formula (HI.), page 374, the strength of a beam of any dimensions, whose form of section is similar to any of those forms, using the word similar here in its strict geometrical sense, as explained in definition 12, page 103. 1. For the rectangular form of section :— From the mean of 265 experiments of Messrs. Hodgkinson and Fairbairn, it appears that a weight of 454*4 lbs. applied at the center of a bar of cast iron 1 inch square and with a bearing of 4*5 feet produced fracture ; therefore, for a bar of any other dimensions, putting W for the breaking weight applied in the center, L the length of the beam in feet, h and d its breadth and depth in inches, we have 2045 6rf' _ . ,^ = W, m lbs. L 18'25^(/^ L •912 b d' = W, in cwts. =s W, in tons. (VIII.) 2. For Mr. Hodgkinson's form of section (fig. 235) in which the *i%a (a) of the lower flange is made six times that of the upper, wc We 4862 ad _ . ,, = W, m lbs. L 48-33 acf L 2166a</ = W, in cwte.J (IX.) = W, in tons. I S8S TRAirSVEBSB 8TBSN0TH OF CAST IRON. [PART II 3. When the form of section does not materially differ from that shown in figure 235, the following formula is found hy Mr. Hodg- kinson to agree moderately well with the results of experiments. If h^ equal the entire hreadth of the bottom flange, b^ the thickness of the vertical part, d^ equal the whole depth of the girder, d^ the depth without the lower flange, all in inches, and L equal the length in feet, we have ^^^^ {^ ^i' - (*i - ^i) ^1 = W, in lbs. "* <L 40 {*i ^i' - (^ - ^2) ^2*} = W, in cwts. > ... (X.) I 4. We have also been favoured with the following arbitrary for- mula by Mr. Dines, which he has found to be tolerably correct in all cases where the length of the girder did not exceed 25 feet, its> depth in the center was not greater than 20 inches, nor the breadth of its bottom flange less thim one-third, nor more than half th&- depth, and the thickness of the metal not less than -ji^th of th^s- depth. Then the letters expressing the same quantities as Hpfnw* we have 170^ "V -j- {«, d," - (i. - b,) dn = W, in lbs. 80 5l7 {*i •'i* - (*, - *,) ''/} = W, in cwU. ... (XI.) ~ {4, rf.» - (J. - h) «/«*} = W, in toBs. LONOrrUDINAL FORM OF BEAM OF UNIFORM STRENGTH. 1. I/the depth (^tke beam is uniform: — When the whole load is collected in one pohit, the sides of th^' beam should be straight lines, the breadth at the ends being half thi^^ where the load is applied, as in fle. 286. When the load is uniformly dntributed, the sides shookl be pos^^ tions of a circle, the radius of which should equal the square of tli^v length of the beam, divided by twice its breadth, as in fig. 237. 2. When the breadth of the beam i» uniform: — When the load is collected in one point, the extended side of ttr^ beam should be straight, the depth at the point where the load ^ applied twice that at we ends, and the linea connecting them straight, as in fig. 238. CHAP. VIII.] TBAN8V£R8B 8TBEN0TH OP MATERIALS. 387 When the load is uniformly distributed, the extended side should be straight, and the compressed side a portion of* circle whose radius equals the square of half the length of the beam divided by its depth, as in fig. 239. 3. When the transverse section rf tie beam is a similar figure tknmghout its whole length: — When the load is collected ib one point, the depth at this point should be to the depth at hm extremities as 3 : 2 ; the sides of the beam being all straignt lines, as in fig. 240. When the load it vniformly distributed, the depth in the center should be to tba depth at the end as 3 : 1 ; the sides of the beam being all stnHJgbt lines, as in fig. 241 . mANSYKBSE STBENOTH OP OTHER MATERIALS THAN CAST IRON. The only form of beam which is employed of any other material than cast-iron is the rectangular form, the strength of which may be immediately obtained from the following formula, in which h is the breadth, d the depth, both in inches, L the length in feet, and W the breaking weight applied in the center ; then ?i^ = W,mlb8. li L o^bd' = W, in cwts. = W, in tons. (XII.) The following table exhibits the values of the constant coefiicients ^^ a,, and a.^ according to Professor Barlow, for several different i^nds of timber. Name of Material. Value of a|. Value of a^ Value of a^ Authority. brought iron Brass, cast ... I«ead, cast •nn,caat Zinc, cast Teak Poon Ash Canadian oak. English oak... I^tch pine..... 2290 890 196 872 746 821 740 676 589 567 544 20-46 7-96 1-76 3-32 6-66 7-83 6-61 603 6-26 4-98 4-86 •022 •397 •087 •166 •338 •366 •380 •301 •263 •249 •243 Tredgold. Barlow. c c 2 388 ELASTICITY OF BODIES. [PABT II. Name of Material Beeclj Dantzic oak Adriatic oak Red pine Mar Forest fir.... New Eugland fir. Riga fir Elm Larcb Value of a,. 519 486 461 447 408 367 359 338 330 Value Valae of a^ of aj. Authority. 4-63 •232 Barlow. 433 •216 jy 412 •206 >1 400 •200 >^ 3-64 •182 99 3-28 •164 99 3-21 •160 99 302 •151 99 2-94 •147 99 THE STRENGTH OF BEAMS ACCORDIXO TO THE MANNER IN WHICH THE LOAD IS DISTRIBUTED. In the same beam, the weight which will be required to break it depends very much upon the point at which it is applied ; and upon, the ^-ay in which it is distributed ; as also upon the manner in whid^ the beam is supported. 1. When the beam is supported at both ends : — If the weight which must be applied at the center to ^ produce fracture, equals ) • Then that which must be applied at any other point, \ (where /j and l^ equal the distances of that point from r L' — each support, and L the distance between the sup- I ^ I I ports,) ii^ill be equal to ; And that which would be required if distributed ) o w uniformly along the beam, would be equal to J 2. When the beam is supported only at one end : — If the weight is applied at the other, it must be ) i t^ . equal to J * ' And if distributed uniformly along the beam, it must ) i »r be equal to J * Sbct. V. Elasticity of Bodies subfected to a transverse strain, The deflection of rectangular beams when supported at each exL - mity and loaded in the middle, is found by Uie following fonnu -^ in which i equals the deflection in inches, Wj the weight in Iba. pr" "^^ ducing it, and m the weight of the modulus of elasticity in lbs. ^^^ the given material, which is given at pages 38? and 384 ; the oUt^^^ letters represent the same quantities as before, tlien we have '=i^^ («"■) CHAP. VIII.] ELASTICITY OP BODIES. 389 If tbe weight be uniformly distributed, instead of being collected in tbe center, it will only produce |th8 of tbe deflection given by tbe above formula. For cast-iron girders whose dimensions are limited, as described in § 4, page 386, Mr. Dines finds that the following formulee may be made use of to determine the deflection of the beam when loaded in the center with |ths of the breaking weight ; in which d is the depth of the beam in the center in inches, and L the length in feet. When the top and bottom flanges are equal, and the girder parallel, or of equal depth throughout — * = i^ .(^^^-^ When the flanges are not equal, and the girder is not parallel — * = ^. (^^-J When the beam has no top flange, and the depth varies, then ^ = 8^^ (^VI-) Thefte formulee are the result of upwards of 2000 experiments upon beams of a moderately large size. In the first section we stated that the elasticity of a cast-iron beam becomes injured with a strain very much smaller than the breaking weight, so that, when the strain is removed, the beam does not recover its former shape, but remains permanently bent. Mr. Hodg- kinson was, we believe, the first writer who noticed that such was the case, which he has proved by his experiments, and which result has been fully confirmed by those of Mr. Cubitt. The greater part of Mr. Hodgkinson's experiments were made apon beams having only a bearing of 4 feet 6 inches ; and he found that the amount of the permanent set varied as the square of the weight applied. This rule, however, although correct for beams about 5 feet in length, does not apply when the length becomes mnch greater, for Mr. Cubitt found by his experiments, that when the length became equal to about 20 feet, the set was only as the weight, and that with larger beams the set was still less. As these exponents are important from the size of the girders, we have tabulated a few of them, as on the following page. 390 MR. GUBITT8 SXPBBIMBNTS. [part II. 1 n n 1 i ij 1 ip 1 BBM^ftKa. L 2 i 8 10 14 18 "000 172 -206 '30fi ■4B0 -iia4 7S2 sa4 ■100 ^ 0070 ■100 Zim '^^ ■006 ,S.m ' ^^^ 102 ^]l mm The form and dimeii«Joni of theae girdezi are ahon*!! in %« 242. The result here gtTen are tbe mean of two gif4et%, one of which weighed 41 cf»t, qn. 20 Ibs.^ Attd the other 41 cwt. 2 qrt. 6 Ibft. The (otU length of the: girders ww 30 fl. 4 in., the kngth of bearing 28 ft. ^ 3 4 10 12 14 IG IB Ofll 121 ■221 305 '300 -ao3 005 '70» 300 oil i^lJS '^'^^ j*0800 '0050 1 Vm% from tUe wmje jmntsni Uj 0050 No.t The rwulta are the moml -0045 uf two girdent, one of which <W70 weighed 42 cwu qn. 14 IbfcJ 0105 the other 41 cwt. 3 qrt. 16 Ibt^ 0105 The length of beftring wma 37 £tM 11165 ?«**^ 1^ 3. 2 4 6 8 10 12 •117 213 '406 096 133 •120 -124 •lie -020 025 •032 04$ •057 -068 ^2 The result* arc the oieAn of two jjri girders, One of which ireighed 26 ^\l Wt. Iqr. 26 lb*,, and the olher ;?{* ,25 cni. 2 qm 22 lb«. The tengtb "* * |of the girders wm 24 ft, 6 In., the ilen^ o( beanng 33 ft 10 in. ^^ 4. % 4 6 B 10 11 125 '374 H03 012 071 '140 ■log 110 110 m9 0150 0245 0350 mm '0620 mm 1 Cam from the wiQe pfttUifn ■« -0095 'Nil. 3. The faults Are the ni«ftn ■0105 of two girdeiTft, one of vhich ■0130 weighed 2&cm, 2 qr«, Iht, uid 0140 the oiher 24 cwt. 3 qn, I7 Ibsk. 0070 The lengtb of the be»riBg vm |23 ft. ft. 4 a 10 13 '24 '38 '50 '64 74 12 -14 14 •10 •035 050 065 •HBO 1 The fonn and d]aie»i»k»ni of tb^^^ -01 ft prd*r ajneihown in hg. 244, Th* ^^ rtiR 1 weight of the giMer wm 10 cwt*^ ^^ ;?{* (0 qr». \B lbs., its total length ^r^ ^*^ Ifeet, and the distftnce betvup. ' ■ [the bcftrLngm 20 ft. 7 iru 6. 1 2 3 4 4i 085 4:4 380 1 402 475 1 -0«9 100 123 073 018 -035 •000 •103 013 ' m& ■000 ; ■017 043 1123 E«ctiuiguliir bar of out ln»=3 51 inches in depth* 1^04 inditf in breadtht and 5 ft, 6 in, lengC- of bearing. Broke at & tons. f 7. 1 3 4 44 083 16! 360 '300 '440 •078 OtfO 120 •060 015 1Ki2 •030 RecL^ngulELr bur of emm iraiiw Bi incliei to depth, I 06 incMr in br^dth, and 5 feel 6 ipcha lengUi of b«mng. Broke v%^ 41 tonA^ CHAP. VIII.] MR. CUBITTS EXPERIMENTS. 391 i d 1^ ^1 11 Q h I*' 1 a mmrABKii ' 1 2 3 4 ft 6 11*7 100 146 105 •250 350 063 045 060 066 100 -005 -009 ■020 ■lao ■004 ■Oil ■160 RecUngulAr tmr of wucmght Iron, 4 '97 mdies in depth, 1 inch in breath, and 5 ft 6 in. length of be&ring. 1^ 1 1 1 S 3 4 ft 6 060 110 166 210 300 630 *^* No tat, 0^ i 000 OfrS 1 012 090 -5541 230 ! -260 ■006 il3a ■2O0 Kecungulfir b^r of wrotight iron, 5 '03 incbeii in depth* 1 IqcIi in breadth. And 6 ft, 6 in* lin^ of bearings 11K S 3 4 G 100 146 •206 402 but -046 060 197 -130 Trifling ■01 a ■154 '280 ■013 130 ■126 Rectangul^ bar of wrought iron, 6 inc&Bi m depth, '97 incheii In breadth, and 6 ft* 6 in. length of bearing. 11. 3 3 4 . ft loe 160 318 310 380 ■062 ■oeo -093 *070 Trkfling ■0O8 ■044 •106 ■012 <>24 il6l Hecuingular hsr of wrought, iron, 4'97 inches In depth, 102 inchefl iti bresfitbi aod 6 It. 6 In, length of bearingp IS, 1 3 3 ' 4 •060 106^ 160 -200 -238 26a ■066 i)46 ■066 -032 ■030 Set. 004 ■008 ■016 ■020 ■028 0O4 ■007 -006 008 Rectangular bar of wrought iron, 4 97 mckim in depth, 1^02 inches in breadth, and 6 fu 6 in. length of bearing. 13. 1 3 3 4 41 060 103 148 •200 -230 -063 •046 ■062 •020 Set ■004 BectJingular bar of wrought iron, 4'»7 *nche« in depth, 102 inchei in breadth, and 6 fU 6 in, length of bearing. 14. I li 3 n 3 Si 4 •162 ■302 '348 ■300 ■374 ma ■060 046 062 074 ■002 ■003 008 ■010 ■066 •306 •001 005 •002 046 ■161 Rectangular bar of wrought iron, 4 Inches in deplh^ 1^01 inches in brewJth^ and 6 f^. 6 in, length of bearing. IS. I i' 4 41 ■100 150 loa -296 '3^ ■460 G&8 4)40 046 •oeo ■OGO ■006 '208 ^103 •010 ■016 ■023 046 102 ■266 mi 006 ■007 ■023 •066 163 Rectangular bar of wrought iron, 4 incbea in depth, I -01 Ini^ei in breAdth, and 6 ft 6 in. length of bearing. 392 STRBNOTH OP MATERIALS. [part II^ Sbct. VI. Strength of MaieriaU to resist Torsion. The power of a bar or rod of any material to resist torsion, maj^ be measured by the angle through which the end of a lever attacbec^ to the same will be moved by a given weight. If D equal the dis^ tance from the fixed end of the bar or prism to the point of applica^i tion of the lever used to twist it, / the length of the lever in inches w the weight in lbs. applied to the end of the lever, r the radius c^ the pnsm if cylindrical, and 5, d^ its breadth and depth if rectangulau4 all in inches, § the angle of torsion at the point of application, F a cona stant for each material, representing the specific resistance to flexur . by torsion, and W a constant weight expressing the resistance to to ^ sion, with regard to a unit of surface at the time of fracture ; th^ the following table will exhibit the relations subsisting between thc^ quantities. Form of section of the prism. Ketistanoe to angular flex- ure by a force of torsion. Resistance to fracture I a force of torsion. Cylindrical . Square .... Rectan":ular F = »/ 2L W F=«/i(f-±^)i^ wl 2_ e The mean value of W for cast iron, as obtained from the experi- ments of Messrs. George Rcnnic, Bramah, and Dunlop, is 32,503 lbs., and from Mr. Bevan's experiments, the mean value of F for cast iron is 5,709,600 lbs., and for wrought iron and steel J 0,674,540 Iba^ and, these values being substituted in the above table, it becomes PonnorMclion of the prism. Cylindrical Square Rectangular Retlftance to angular flexure by a force of tornoo. . Cast iroD. u»/ -- 8968620— . tr/-96I600 d*0 v/-» 1903200 b*dU (6« + rf«)L Wrought iron or steeL ir/= 16767770—. d*i ^/- 1779090—-. Rcristance to ftactwc' aforceoftontoa. Cattfaon. w/- 51066 r«. w/»76Clif*. IV/-108S4- APPENDIX. TABLE OL [Ho, Um h. 175802 t s 4 5 « 7 S 1 « N. ISl ISQl 8876 llHO B*2U 1734 8038 2188 6468 2598 8894 3029 7321 3461 7748 3S01 8174 100 1 2 3 4 6 8 7 8 9 no 1 2 8 4 8 7 8 9 120 1 2 3 4 5 8 7i 8 9 ISO 1 2 3 4 5 6 7 3 140 1 2 3 4 5 6 7 8 m 0300 4521 8700 0724 4940 9116 1147 5360 0532 1670 5779 0047 1993 6197 2415 8616 \m 0361 4488 8571 0775 4896 S078 m 242S 6533 2841 6042 ' 3252 7350 3664 7757 4075 8164 1^5 *S7 060O 462S 8620 3 1004 5029 0017 4 1408 5430 9414 5 18)2 5830 9811 6 2218 1230 2619 6629 3021 7028 £23 8 1 0207 7 4148 8053 0602 8 4540 8442 0098 4038 8830 M M 2578 6495 03S0 4^230 3048 2989 6885 3362 7275 8755 7664 {46 0768 4613 8428 1153 4006 8805 1538 5378 9185 1024 5760 0563 2309 6142 0042 2694 6524 ' M»ti 0320 4088 7816 152 106 1829 5580 9298 2206 5953 9658 2582 8326 2058 8609 3333 7071 3709 7448 m^ 0038 3718 7363 5 0407 4085 7731 6 0776 4451 8004 7 1 1145 4818 8457 8 1514 5182 8819 m rre 8 29B5 6640 3 3352 7004 4 m 0206 3881 7426 0826 4210 7781 0087 4576 8138 1347 4034 8400 1707 5201 8848 2067 5647 0198 2428 6004 0552 ' Ul 22 0983 4471 7051 1315 4820 8208 1667 5160 8644 2018 5518 8990 2370 6866 9335 2721 6215 9681 3071 6562 HI4 0028 3462 6871 1403 4828 8227 1747 5180 8585 2001 5510 8903 2434 5851 0241 2777 6191 9570 B119 6531 9018 m» 0253 3609 6040 :63 8 in 1509 8 4644 8265 1934 4 5278 8596 2270 5 5811 8926 2605 6 5943 9256 2940 7 6276 9586 3275 B 6608 9915 m 0245 3525 6781 mi m 1560 4830 8076 12&8 4406 7671 1888 5156 8300 2216 5481 8722 2544 5S06 9045 2871 8131 9368 8198 6456 9600 £»3 0012 3219 6403 9564 ,77 77 1819 4814 7087 1030 5133 8303 2260 5451 8618 2580 5760 8034 2000 8086 9249 m m 48 0822 3051 3 7058 1136 4263 4 736T 1450 4574 6 T676 1763 4835 6 7085 2076 5196 7 8204 2380 5507 8 8603 2702 6818 9 8911 m »43 0142 3205 6246 9266 0440 3510 8549 9567 0756 3815 8852 9868 1063 4120 7154 1370 4424 7457 1676 4728 7750 1082 6032 8061 »66 0168 3161 6134 9086 0489 3480 6430 9S80 0769 3753 6728 0874 1068 4055 7022 0988 m 'OS 2266 5244 8203 2564 6541 8497 2863 58S8 8792 148 eft IHl 4080 H34 4351 1726 i 4641 2019 14932 2311 2803 6512 2696 ; 5803 *B K(>. l&OO L. 170091] TABLB III. [N«. 1999 L. 30031 1 m 1 s 3 * 9 160 1 , 2 a 4 1 5 1 2 3 4 S 6 r 1 a e s 176091 6381 6670 0652 6969 9839 724S 7536 7825 8113 8401 8689 15 8977 i*ao* 0126 2985 5825 8647 ' 0413 3270 0103 8928 0609 35^5 6391 1 0209 0986 38^9 6674 9490 1272 4123 6956 9771 1568 4407 7230 1; 2 S 4 5 « 7 SOD 1 2 1 S ! i 1 » 7* 1 1 a' 1S1S44 4 SOI 7521 2129 4975 7803 2415 6259 80S4 2700 1 5542 8366 msi 2346 5623 £382 190332 3125 8667 0612 3403 6170 0892 6453 1171 3959 6720 H81 1461 4237 7005 9755 1730 4514 7281 2010 4792 7556 2289 : 2667 6009 1 5346 7832 8107 8932 9200 0029 2701 6 6475 8173 0303 3033 6 5746 3141 0577 ; 0S5O 11^4 3346 f .^' 924TI a*l397 4120 6326 16T0 1 4391 7096 1»43 2 466S 7366 2216 % 4934 7634 2488 4 6204 7004 3305 7 6016 8710 ^77 8 6236 3979 9515] Ui&a 0061 1*319 2980 5638 8273 0586 3252 5902 8530 0353 3518 6160 3793 1121 3783 6430 9U0O 1368 4049 6694 0323 1054 4314 6957 9535 IU21 ! 45711 '' 7221 9845 212lB8i 2464 4844' SI 09 7484 7747 2720 5373 8010 ! 6 7 S ,220108 0370 0631 1 2716 2970 \ 3236 £J09 5503 1 5826 0892 3490 0U84 8667 1153 3755 6342 3913 1414 4U15 6000 9170 1075 4274 6858 9420 193«j 4533 7115 9682 2190 4792 7372 0938 245ff 5051 76S0 9 m 1 3 7887 1 8144 1 8100 am . 1 ' 2742 1 5276 7793 , 1 230440 0704 2996; 3250 £528 5781 2 OOtiO 3604 6U33 8548 3 1215 3767 6285 8799 4 1470 4011 6537 9049 5 1724 4264 6789 9299 1979 4517 7U41 1 9550 7 2234 4770 7292 0800 i 243S 5023 7544 8046' 8297 0050 2541 5lM9 7482 9032 om 4 e 7 240549' 0799 3038 3280 6513 6759 7^73 8219 1U48 3534 60U6 8464 1297 3782 6252 8709 1540 4030 6499 8954 1795 4277 6745 0198 2044 4525 6991 9443 2293 4772 7237 9087 27*6 1 5M J 8 ISO 1 25U420 2853 < 5273 7679 U604 3096 I &614 7918 0^08 S338 2 5756 8158 1161 3580 3 6090 3398 1395 3322 4 6237 8637 1638 4064 6 6477 8877 1081 4306 6 0713 9116 2l:£5 i 2:163 4548 1 4790 7 ! « 6963 7193 9355 9594 20l«) 5031 9 7431? a m I . t i 260071 2461 4818 7172 0613 0310 0548 2688 1 2925 6U64 52^0 7400 7641 9740 9080 0787 3162 5525 7876 1025 3399 5761 3110 1263 3630 5990 3344 1601 3873 6232 8678 1739 1970 4109 4340 6467 6702 3812 1 9040 1214 0213 2538 4350 7151 3 9439 iHl46 2770 50B1 7330 ; 4 9067 Otfr9 3001 5311 7609 5 9895 0912 3233 5542 7838 6 1144 3464 5772 8067 7 1377 3606 0002 i S296 8 im V 8 9 271842 4158 6163 2074 4389 0692 1 2ay(J 46:i0 0921 2 39iT 190 8754 8932 ; 9211 0123 2306 4056 6905 9ua 0351 2022 4832 71 SO 9366 0578 1 2849 6107 T354 9539 "«Wi m 1 t 3 2810331 1201 ' 1488 3301 3527 ^753 5657| 5782 6007 7S02I 8023 8249 1716 3979 6232 8473 1942 4205 6456 8696 2169 4431 6631 3020 m m m 1 i i i 1 i 1 5 7 S 9 290OJJ5 2266, 4400 6666 8863, 0267 2478 4687 68^4 i 0071 0480 mn 4907 7104 0289 0702 2920 5127 1 7323 9507 0925 S141 6347 7542 9725 1147 3363 5567 7761 0043 1309 3584 5787 7979 1591 3304 6007 3198 1313 4026 0226 8416 i 1 t 0161 0373 0696 1 oau 1 V%i0io 1^0010301 TABLB IIL [N*. 2499 J,. 897766 1 1 2 3 4 6 1 T & 9 1 S 8 4 6 6 7 8 9 K. 301030 S19@ 7490 M30 311754 S3&7 5^0 1247 8412 8566 7710 0843 wm 4078 6180 8S72 1464 3628 8781 7924 1681 3844 5996 8137 1398 4069 6211 8361 2114' 4275 6425 8564 2331 4491 6639 8778 2647 4706 6854 8601 2764 4921 7068 9204 2980 5136 7282 9417 2oa: li 2 3 4 5 6 7 8 6 0066 2177 4239 6390 8481 0263 2380 , 4490 6699 8689 0481 2600 4710 6809 8808 0693 2812 4020 7018 9106 0906 3023 5130 7227 6314 1118 3234 5340 7436 9522 1330 3446 5561 7646 6730 1542 3656 5760 7854 9938 320146 0354 05^2 j 0769 0977 1184 1391 1593 1805 2012 1 2 3 1 5 ! d, r 2210 4282 6336 1 2426 4488 6541 2 2633 4694 6746 8787 3 2839 4869 6950 8991 4 3046 5105 7155 9164 5 3252 5310 7359 9308 6 3458 5516 7563 9601 7 3665 5721 7767 0806 8 3871 5926 7972 9 4077 8181 8176 210 1 2 l^3S0 nsh^ 0008 2034 4051 6059 8058 0211 2236 4263 6260 8267 3 330414 oeir ; 2438 S64(» 4154 16^ 6460 6660 84^6 86M 0819 2842 4856 6860 S855 1022 3044 6057 roflo 9064 1226 3246 6257 7260 9253 1427 3447 6458 7459 9451 1630 3649 6658 7659 6650 1832 8860 5869 7866 9849 41 6 6, T U047 2028 0246 fl S40444 064^ 0841 1030 1237 1436 1632 1830 2226 9 1 s 1 ! 2423 43&2 e353 S30j^ 1 2820 4689 6549 8600 2 2817 4786 6744 8094 3 3014 4081 6039 8SS9 1 3212 5178 7135 9083 5 3409 6374 7330 9278 6 3606 5570 7525 0472 7 3802 5766 7720 9666 8 3969 6662 7615 9860 9 4196 2^0 6167 1 1 8110 1 a 0064 3 350-^48 218S 4108 eo2« 7935 »835 0442 S375 4301 8217 81 35 0636 2668 4493 6403 8316 0829 2761 4686 6699 8606 1023 2654 4876 6790 8696 1216 3147 6068 6981 8886 1410 3339 6260 7172 6076 16i>3 3632 5462 7363 9266 1708 3724 5643 7554 9456 1689 3910 5834 7744 9646 4; 6 6 7 8 9 oose 021^ 040^ 0593 0783 0972 1161 1350 1539 L 1 3 3 1 301728 1»17 M12: 8800 W88' fi675 736« 7543 9218 9401 210# 8938 6862 77^ 9687 3 2294 4176 8049 7015 ©772 4 2482 4363 6236 8101 9658 6 2871 4551 6423 8287 0143 1091 3831 5664 7488 9808 6 2859 4739 6610 8473 0328 2176 4016 5846 7670 9487 7 3048 4626 6796 8659 0513 2360 419B 60-26 7862 9668 8 3238 5113 6083 8845 0698 2544 438? 6212 mu 9849 6301 7169 6030 08S3 330 I 3 4 • 37l0fiS 1263 2912' 3096 4748 4932 8677, 6769 8S98' 8630 1437 3280 5115 6942 8761 1622 3484 5298 7124 8943 JbOe 3647 5481 7308 9124 2728 4566 6364 8216 5 8 9 0030 9 1837 3636 5428 7212 8989 9 240 1 S 8 4 5 6 r 8 9 »1T 8815 M06 r»90 i 038^ 2197 8995 8785 75$8 0673 2877 4174 5964 7746 9620 3 0754 2557 4363 6142 7924 9698 4 0634 2737 4533 6821 8101 0876 5 1115 2917 4712 6499 8279 6 1296 3097 4891 6677 8456 7 147.6 3277 5070 6356 8634 8 1666 3456 6249 7034 8811 \ ' 9166 9348 0051 1817 3575 6326 7071 0223 1993 3761 5501 7245 0406 2166 3936 0582 2345 4101 6850 7592 0769 2521 4277 6026 7766 y 39093^ 1112 i8»7 2878 4452 4627 Bim 08T4 1238 3048 4802 6^8 1464 8224 4977 ! 67i2 1841 3400 6152 6396 *Ba No. 2600 L. 397940] TABLB III. [No. 2999 L. 476976 8461 I 8634 2 401401 8 3121 4 4834 5 6540 6 8240 7 9933 6710 I 6881 8410 8579 0365 2089 3807 5617 7051 I 7221 8749 ' 8918 8808 , 8981 9154 0538 0711 2261 2433 3978 I 4149 5688 5858 7391 I 7561 9087 ! 9257 8 411620 9, 3300 260 4973 1 6641 2 8301 8 9956 4 421604 5 3246 6 4882 7 6511 8 8135 9 9752 0102 0271 0440 ' 0609 ! 0777 0946 1788 j 1956 I 2124 2293 2461 ' 2629 3467 3035 , 3803 3970 4137 ! 4305 1 5140 6807 i 8467 ' I 6 2 ' 3 4 5307 6474 | 5641 6808 , 5974 6973 7139 , 7306 I 7472 ' 7638 8633 8798 8964 , 9129 9295 0883 2605 4320 6029 7731 'N. 9328 ' 9501 250 1 1056 , 1228 2777 2949 4492 4663 6199 ' 6370 7901 8070 9426 9595 9764 1114 1283 2796 2964 1451 3132 4472 I 4639 ' 4S06 ] 9^ 8 6141 ' 6308 , 6474 260 7804 i 7970 , 8135 9460 9625 ' 9791 1 0121 1768 3410 6045 6674 8297 9914 0286 0451 0616 1933 3574 5208 2097 I 2261 3737 3901 6371 5534 6836 I 6999 | 7161 8459 8621 8783 0075 j 0236 i 0398 270 431364' 1 2969 2 4569! 3 6163 7751 9333 4 6 6 440909 7 2480 8 4045 9, 5604 280 7168 1 8706 1 1526 3130 4729 6322 7909 9491 1685 3290 4888 6481 8067 9648 8 1846 3450 6048 6640 8226 9806 4 2007 3610 5207 6799 8384 9964 1066 1224 1381 1538 2637 2793 2950 3106 4201 4357 4613 4669 6760 6915 6071 6226 1 2 8 4 7818 7468 7623 7778 8861 9015 9170 9324 2 450249 8, 1786 41 3318 6 4845 6 6366 71 7882 8 9392 0865 , 2400 I 3930 ! 5454 6973 , 8487 9995 9 460898 290 2898 li 8893 2 6883 8 6868 41 8847 6 9822 0657 0704 2026 2171 8487 8638 4944 6090 6897 6542 . 8000 L. 477121]^ TABLE III. [No. 3499 L. 543944 1 2 3 4 5 6 7989 7 8 1 9 N. 800 ) 477121 7266 7411 7555 7700 7844 8133 8278 1 8422 L 8566 8711 8855 8999 9143 9287 1 9431 9575 9719 9863 1 2 I 480007 0151 0294 0438 0582 0725 0869 1012 1166 1299 \ 1448 1586 1729 1872 2016 2159 2302 2446 2688 2731 3 I 2874 3016 3159 3302 8445 3587 3730 3872 4016 ! 4157 4 > 4300 4442 4585 4727 4869 5011 6163 6296 5437 6679 5 5 5721 5863 6005 6147 6289 6430 6572 i 6714 6866 6997 6 ' 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 7 \ 8551 8692 8833 8974 9114 9255 9396 9637 9677 9818 8 9 > 9958 0099 1 0239 2 0380 3 0620 4 0661 5 0801 6 0941 7 1081 8 1222 9 ) 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 310 2760 2900 3040 8179 3319 3458 3697 3737 3876 4015 1 I 4155 4294 4433 4572 4711 4860 4989 5128 6267 5406 2 I 5544 5683 5822 5960 6099 6238 6376 6516 6663 6791 8 ^ 6930 7068 7206 7344 7483 7621 7759 7897 8036 8173 4 i 8311 1 9687 8448 9824 8586 9962 8724 8862. 8999 9137 9275 9412 9550 5 6 7 0099 1470 0286 1607 0874 1744 0611 1 0648 0785 2164 0922 2291 501059 1196 1333 1880 2017 2427 2564 2700 2837 2973 3109 3246 3382 3618 3656 8 8791 3927 4063 4199 4335 4471 4607 4743 4878 6014 9 1 2 3 4 5 6 7 8 9 5150 5286 5421 5557 5693 5828 5964 6099 6284 6370 820 6505 6640 6776 6911 7046 7181 7316 7461 7686 7721 1 7856 7991 8126 8260 8395 8530 8664 8799 8984 9068 2 9203 9337 9471 9606 9740 9874 0009 1349 0148 1482 0277 1616 0411 1760 3 4 510545 0679 0813 0947 1081 1215 i 1883 2017 2151 2284 2418 2561 2684 2818 2961 3084 5 1 8218 3351 3484 3617 3750 3888 4016 4149 4282 4415 6 ' 4548 4681 4813 4946 5079 5211 5344 6476 6609 5741 7 ( 5874 6006 6139 6271 6403 6535 6668 6800 6932 7064 8 » n96 7328 7460 7592 7724 7855 7987 8119 8261 8382 9 1 2 3 4 5 6 7 8 9 > 8514 I 9828 8646 9959 8777 8909 9040 9171 9303 9484 9666 9697 330 1 2 0090 1400 0221 1530 0353 1661 0485 1792 0615 1922 0745 2053 0876 2183 1007 2814 S 521188: 1269 \ 2444 2575 2705 2835 2966 3096 3226 3366 3486 3616 3 I 8746 3876 4006 4136 4266 4396 4526 4666 4785 4915 4 > 5045 5174 5304 5434 5563 5693 5822 6961 6081 6210 5 S 6339 6469 6598 6727 6856 6985 7114 7243 7372 7601 6 r 7680 7759 7888 8016 8145 8274 8402 8531 8660 8788 7 J 8917 9045 9174 9302 9430 9569 9687 9815 9943 0072 1361 8 9 > 530200 0328 0456 0584 0712 0840 0968 1096 1223 1 2 3 4 5 6 7 8 9 ) 1479 1607 1734 1862 1990 2117 2245 2372 2500 2627 340 I 2754 2882 3009 3136 3264 3391 8518 3645 3772 3899 1 I 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 2 ) 5294 5421 . 5547 5674 5800 5927 6063 6179 6806 6432 3 1 6558 6685 6811 6937 7063 7189 7316 7441 7667 7693 4 S 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 5 Jl^9076 9202 9327 9452 9578 9703 9829 9954 0079 1830 0204 1454 6 7 r 540329 0456 0580 0705 0830 0955 1080 1205 Bl 1679 1704 1829 1953 2078 2203 2827 2452 2576 2701 8 r. 2820 8950 3074 8199 3323 3447 8571 3696 9820 8944 9 No. 8500 L. 544068] TABLE III. [Xo. 8999 L. 601951 N. 8 850,544068 1 6307 2 6548 8 7775 9003 4192 I 4816 5431 I 5555 6666 6789 7898 ' 8021 9126 9249 4440 5678 6913 8144 9371 4 5 6 7 8 4564 4688 4812 4986 5060 5802 5925 6049 6172 6296 7086 7159 • 7282 7405 7529 8267 8389 8512 8635 8758 9494 9616 9739 9861 99S4 9 S. 5188 350 6419 1 7652 &SS1 550228 1450 2668 3883 5094 860 1 2 8 4 5 6 0351 0478 1572 , 1694 2790 ' 2911 4004 4126 5215 5836 ! 1 6303 6423 7507 7627 8709 8829 9907 "0026" 2 6544 7748 8948 0595 1816 3038 4247 5457 3 6664 7868 9068 0717 1938 3155 4868 5578 4 6785 7988 9188 0840 2060 3276 4489 I 5699 I I 5 6905 8108 9308 , 0962 I 2181 I 3398 I 4610 > 5820 i 6 7026 8228 9428 1084 2808 3519 4781 5940 7 7146 8349 9543 1206 2425 3640 4852 6061 01i« 1828 2547 8762 4978 6182 561101 1221 2298 2412 3481 3600 4666 4784 8 5848 5966 9 7026 7144 870 ! 1 I 1 8202 8319 9874 9491 0146 1340 2531 3718 4903 6084 7262 2 8486 9608 0265 1459 2650 8837 5021 6202 7379 3 8554 I 9725 0385 1578 2769 8955 5139 6320 7497 4 8671 9842 2 570543 0660 S80 1709 1825 2872 2988 4031 4147 5188 5308 6341 6457 7492 7607 8639 8754 9784 1 9898 580925 2063 3199 4331 5461 6587 7711 8882 9950 0776 1942 3104 4263 5419 6572 7722 8868 2 0893 2058 3220 4379 5534 6687 7886 8983 1039 2177 I 3312 4444 , 5574 6700 7823 8944 0012 I 1153 1 2291 I 0061 I 591065 2177 8286 4393 5496 6597 7696 8791 9883 890 1 2 8 4 5 6 7 8 . 9 e mfsj^^ 1 1176 2288 3897 4508 5606 6707 7806 8900 9992 3426 4557 5686 6812 7985 9056 0173 2 1287 2399 3508 4614 5717 6817 7914 9009 0126 1267 2404 3539 4670 5799 6925 8047 9167 1010 2174 8336 4494 5650 6802 7951 9097 0504 1698 2887 4074 5257 6437 7614 5 8788 9959 0624 , 0743 1817 I 1986 ' 3006 i 4192 I 5376 6555 ' 7732 1126 2291 . 3452 ; 4610 ! 5765 6917 8066 9212 0101 0284 3 1399 2510 8618 4724 5827 6927 8024 9119 0210 0241 I 0855 1381 ! 1495 2518 2631 3652 3765 4783 4896 5912 i 6024 7037 ; 7149 8160 I 8272 9279 1 9391 3125 4811 5494 6678 7849 6 7 8905 9023 8 9 7267 ' 7337 $60. 8469 ! 8589 I 9667 ■ 9787 2 S 4 5 6 0863 0982 2055 ; 2174 3244 , 3362 4429 4548 5612 ; 5780 6791 6909 7967 8084 8 9 9140 9257 i70i 0076 1248 2407 3568 4726 5880 7032 8181 9326 , 0193 I 1359 i 2523 ] 3684 4841 5996 : 7147 '■ 8295 . 9441 03U9 i 0426 1. i 1476 15H2 2 I 2639 I 2755 3 3800 495*7 6111 7262 8410 9555 8 0469 1608 2745 3879 5009 6137 7262 8384 9508 0583 1722 2858 3992 5122 6250 7374 8496 9615 0697 , 1836 I 2972 ' 4105 ! 5235 6362 I 7486 ; 8608 ; 9726 ; 3919 i 4 6072 5 6226 «; 7877 "; 8625; f 9^ ^^ i 9 J I 0811 .8W: 1960 li 4218 5| 5848 4 6475 5; 7599 5 8720 '; 9888' 9 0396 0607 4 1510 2621 8729 4834 5937 7037 8134 9228^ 0819 1408 6 1621 2732 3840 4946 6047 7146 8248 9337 0428 16U 0619 6 1732 2848 8950 5056 6157 7256 8868 9446 0687 0730 7 1848 2954 4061 6165 6267 7366 8462 9566 0646 1734 0842 8 1955 3064 4171 5276 6877 7476 8672 iMAK VvOv 0958 I 9| 9 2066 m 8176 1 42S2 3: 6487! 4 7586= 5 mi , «< 9774! 0766 l«4k 0604 1961 4 " iNoi 4000 L. TABLE IIL [No. 4499 L. 653116 N. 400 602060 2169 1. 8144 j 8258 4226- 4834 5805 5413 68811 6489 7455! 7562 8526 8688 95941 9701 8 610660 0767 9 1728 1829 410 1 2 8 4 5 6 2784 8842 4897 5950 70001 8048' 9098 1 2890 8947 5008 6055 7105 8158 9198 2277 8861 4442 5521 6596 7669 8740 9808 0878 1986 2 2996 4058 5108 6160 7210 8257 9802 7 620186 0240 8 1176 1280 9 2214 2818 420 1 2 8 4 5 6 7 8 480 ii 4 5. 6' 1 8249 8358 4282 4885 5812! 5415 6840 6448 7366; 7468 8889; 8491 9410 9512 0844 1884 2421 2 8456 4488 5518 6546 7571 8598 9618 680428, 0580 1444 1545 2457! 2559 1 8468 8569 4477; 4578 5484! 5584 6488- 6588 7490 7590 8489' 8589 9486 9586 0681 1647 2660 2 8670 4679 5685 6688 7690 8689 9686 7i640481, 0581 8: 1474! 1573 9I 2465 2568 440= li 2! 8 4 5 6 76508^ I 1 8453- 3551 4439 4537 5422, 5521 6404' 6502 7883, 7481 8360 8458 98851 9482 1S78 ffi46 0405 1375 0680 1672 2662 2 8650 4686 5619 6600 7579 8555 9530 0502 1472 2440 2886 8469 4550 5628 6704 7777 8847 9914 2494 8677 4658 5786 6811 7884 8954 I 0979 2042 8 3102 4159 5218 6265 7815 8362 9406 0021 1086 2148 4 8207 4264 5319 6870 7420 8466 9511 0448 1488 2525 0552 1592 2628 3 4 3559 3663 4591 I 4695 5621 j 5724 6648 7673 8695 9715 6761 7775 8797 9817 0783 0886 1748 1849 2761 2862 8 8771 4779 5785 6789 7790 8789 9785 4 3872 4880 5886 6889 7890 8888 9885 0779 1771 2761 0879 1871 2860 3 4 8749 8847 4784 I 4882 5717 ' 5816 6698 6796 7676 . 7774 8658 ; 8750 9627 I 9724 2608 8686 4766 5844 6919 7991 9061 2711 8794 4874 5951 7026 8098 9167 0128 1192 2254 5 8813 4870 5424 6476 7525 8571 9615 0284 1298 2860 6 8419 4475 6629 6681 7629 8676 9719 0656 1695 2782 5 8766 4798 5827 6853 7878 8900 9919 0760 1799 2835 6 3869 4901 6929 6966 7980 9002 0936 1951 2968 5 8973 4981 5986 6989 7990 8988 9984 I-QOST 0021 1038 2062 3064 6 4074 5081 6087 7089 8090 9088 0978 I 1077 1970 2069 2969 3058 0599 1569 2536 0696 1666 2633 5 3946 4931 6913 6894 7872 8848 9821 0793 1762 2730 6 4044 6029 6011 6992 7969 8945 9919 0890 1859 2826 2819 3902 4982 6069 7133 8205 9274 2928 , 3036 4010 , 4118 5089 ! 6197 6166 6274 7241 7848 8312 8419 9881 9488 0341 1406 2466 7 8625 4681 6634 6686 7734 8780 9824 0447 1611 2672 8 8630 46S6 6740 6790 7839 8884 9928 0664 1617 2678 9 3736 4792 6846 6896 7943 8989 0864 0968 1903 2007 2939 3042 7 8 3973 4076 6004 6107 6032 6185 7068 7161 8082 8185 9104 9206 0032 1072 2110 3146 9 4179 6210 6238 7263 8287 9308 0123 1139 2153 3166 7 4175 5182 6187 7189 8190 9188 0183 1177 2168 8166 4143 6127 6110 7089 8067 9048 0224 1241 2255 3266 8 4276 6283 6287 7290 8290 9287 0326 1842 2366 3367 9 4376 5383 6388 7390 8389 9387 0283 1276 2267 3265 I 0016 0987 1956 2923 8 4242 5226 6208 7187 8165 9140 0382 1376 2306 3354 9 4340 6324 6306 7286 8262 9237 N. 400 1 2 3 4 6 6 7 8 9 410 1 2 3 4 5 6 7 8 9 420 1 2 3 4 5 6 7 8 9 0118 1084 2053 8019 0210 1181 2150 8116 430 1 2 3 4 5 6 7 8 9 440 1 2 3 4 5 6 7 8 9 No. 4500 L. 053218] TABLE III. [No. 4999 L 69888 N. 1 1 2 3 4 5 6 7 8 , 9 jir 450 653213 8309 3405 3502 3598 8695 3791 3888 8984 4080 4C 1 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 ; 2 5188 5235 5331 5427 5528 5619 5716 5810 5906 6002 ! 8 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 4 7066 7152 7247 7843 7438 7534 7629 7725 7820 , 7916 . 5 8011 8107 8202 8298 8893 8488 8684 8679 8774 ' 8870 | 6 8965 9060 9155 9250 9346 9441 9686 9631 9726 9821 7; 9916 0011 0106 0201 0296 0391 0486 0581 0676 0771 ! 8 660865 0960 1055 1160 1245 1339 1434 1629 1628 1718 9 1813 1907 2002 2096 2191 2286 2380 2476 2669 2663 1 2 8 4 6 6 7 8 < 9 460 2758 2852 2947 8041 3135 3230 3324 3418 8612 3607 m 1 3701 3795 3889 8983 4078 4172 4266 4860 4464 4648 2 4642 4736 4830 4924 5018 6112 5206 5299 6893 6487 3 5581 5675 5769 5862 5966 6060 6143 6237 6331 I 6424 4 6518 6612 6705 6799 6892 6986 7079 7173 7266 7860 5 7458 7546 7640 7733 7826 7920 8018 8106 8199 8298 6 8886 8479 8572 8665 8759 8862 8945 9088 9131 9224 • 7 9317 9410 9503 9596 9689 9782 9875 9967 0060 : 0153 0988 ■ 1080 ! 8 670246 0839 0431 0524 0617 0710 0802 0895 9 1173 1265 1358 1461 1543 1686 1728 1821 1913 , 2005 1 2 8 4 5 6 7 8 1 9 470 2098 2190 2283 2875 2467 2560 2652 2744 2836 2929 4: 1 3021 3118 3205 8297 3390 8482 3674 3666 3758 3850 1 2 3942 4034 4126 4218 4810 4402 4494 4586 4677 , 4769 ' 3 4861 4958 5045 5137 5228 5320 5412 5503 5695 1 5687 j 4 5778 5870 5962 6053 6145 6236 6328 6419 6611 1 6602 1 5 6694 6785 6876 6968 7059 7161 7242 7338 7424 ! 7516 ' 6 7607 7698 7789 7881 7972 8068 8154 8245 8336 8427, 7! 8518 8609 8700 8791 8882 8978 9064 9155 9246 9357 8! 9428 9519 9610 9700 9791 9882 9973 9680336 0426 0517 0607 0698 0789 0879 0068 0970 0154 0246 1060 1151 ; ! 1 1 2 3 4 5 6 7 8 9 i 480 1241 1832 1422 1513 1603 1693 1784 1874 1964 2055 48> 1 2145 2285 2826 2416 2506 2696 2686 2777 2867 2957 2 3047 3137 8227 3317 3407 3497 3587 3677 8767 3857 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 4: 4845 4935 5025 5114 5204 6294 5383 6478 5563 5652: 5 5742 5881 5921 6010 6100 6189 6279 6368 6458 6547 1 6 6686 6726 6815 6904 6994 7083 7172 7261 7351 7440; 7i 7529 7618 7707 7796 7886 7975 8064 8153 8242 8831 1 8, 8420 8509 8598 8687 8776 8865 8953 9042 9181 9220' 9 9809 9898 9486 9575 9664 9753 9841 9930 J 1 2 8 4 5 6 7 0019 8 0107 9 490 690196 0285 0873 0462 0550 0639 0728 0816 0905 0993 49 1' 1081 1170 1258 1847 1485 1524 1612 1700 1789 1877 2 1965 2053 2142 2230 2318 2406 2494 2588 2671 2759 3 2847 2935 3023 3111 3199 3287 3375 8468 3551 3639 4 8727 3815 3903 8991 4078 4166 4254 4842 4430 4517 5 4605 4693 4781 4868 4956 5044 5131 5219 5307 5394 6 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 7 6356 6444 6581 6618 6706 6798 6880 6968 7055 7141 8 7229 7817 7404 7491 7578 7665 7752 7889 7926 8014 1 9 8IOO: 8188 8275 8362 8449 8585 8622 8709 A7M 8888 1 N. 6000L. 698970] TABLE 111. [No. 5499 L. 740284 -•1 500 1 1 2 3 4 5 6 7 8 9 N. 698970 9838 9057 9924 9144 9231 9817 9404 9491 9578 9664 9751 500 1 2 0011 0877 0098 0968 0184 1050 0271 1136 0358 1222 0444 1309 0531 1395 0617 1482 2 700704 0790 3 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 3 4 2431 2517 2603 2689 2776 2861 2947 3033 3119 3205 4 5 8291 3877 8463 3549 3635 3721 3807 3893 3979 4065 6 6 4151 4236 4822 4408 4494 4579 4665 4751 4837 4922 6 7 5008 5094 5179 5265 5350 5436 5522 5607 5693 6778 7 8 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 8 9; 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 9 1 1 2 3 4 5 6 7 8 9 510 7570 7655 7740 7826 7911 7996 8081 8166 8251 8386 510 1 8421 8506 9355 8591 9440 8676 9524 8761 9609 8846 8931 9015 9100 9185 1 2 3 9 AIM 9694 9779 9863 9948 0033 0879 2 3 710117 0202 0287 0371 0456 0540 0625 0710 0794 4 0963 1048 1132 1217 1301 1885 1470 1554 1639 1728 4 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 6 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 6 , 7 3491 3575 3659 8742 3826 3910 3994 4078 4162 4246 7 1 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 6084 8 9 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 9 1 i 1 2 3 4 5 6 7 8 9 '520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 620 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 1 2 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 2 3. 8502 A nooi 8585 9414 8668 9497 8751 9580 8884 9663 8917 9000 9083 9165 9248 3 4 5 voox 9745 9828 9911 9994 0077 0903 4 5 720159 0242 0325 0407 0490 0573 0655 0738 0821 6 0986 1068 1151 1233 1316 1898 1481 1563 1646 1728 6 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 7 8 2634 2716 2798 2881 2963 8045 3127 3209 3291 3374 8 9 3456 3538 3620 8702 3784 3866 3948 4030 4112 4194 9 I ' 1 2 3 4 5 6 7 8 9 530 4276 4358 4440 4522 4604 4685 4767 4849 4981 6013 680 / !< 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 1 . 2 5912 / 81 6727 5993 6075 6156 6238 6320 6401 6483 6564 6646 2 6809 6890 6972 7053 7134 7216 7297 7379 7460 3 41 7541 7628 7704 7786 7866 7948 8029 8110 8191 8278 4 6. 8354 8485 8516 8597 8678 8759 8841 8922 9003 9084 6 6 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 6 7 8 7 8 vv/« 0055 0868 0136 0944 0217 1024 0298 1106 0378 1186 0459 1266 0540 1347 0621 1428 0702 1508 780782 9, 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 9 ^ 1 640 28M 1 2 3 4 5 6 7 8 9 2474 2555 2685 2715 2796 2876 2956 3037 3117 640 1; 8197 8278 8858 3488 3518 3598 3679 3759 3839 3919 1 2 8999 4079 4160 4240 4820 4400 4480 4560 4640 4720 2 ' 4800 4880 4960 5040 5120 5199 5279 5359 5439 5519 3 \ * 5599 6679 5759 5S38 5918 5993 6078 6157 6237 6317 4 5 6897 6476 6556 6635 6715 6795 6874 6954 7034 7113 6 \ 2 7198 7272 7852 7431 7511 7590 7670 7749 7829 7908 6 I 7987 8067 8146 8225 8305 8384 8468 8543 8622 8701 7 \i 8781 9572 8860 9661 8939 9731 9018 9810 9097 9177 9256 9386 9414 9493 8 9889 9968 0047 0126 0205 1 0284 9 10 TA5L1 IIL [ICo. 6999 L. 778079! 8 9 N. -Mai » -t*!^ -s-sti :^.L. .n*.*. * I ! ri. l^i! l^i-ir !!;<* 1 "! s?!.* i' '. * ? Vf il"r 3 STiH i>'-§ ir*?i iKT ♦ *j: •.;=-• 7.^- frii ■ -is! -.r: ♦44.- 45e* f :. -: 5:r.? 52?: «:•> " !.•;.: ;:*•?:? f» :: */•■*> i" -^r:*-* f~:i f-j.. 5?«* > r*:i T**^ r.v~ r*«; 1 ?iH3 >■*• •■?*? >'.i* >*>! 'IT 4 1»44 4-f.fl *i45 5 l<*i4 2411 SIM 4r«2 «323 TlOl T?:3 6 S653 9427 0915 1703 2439 3275 4053 4S40 5«21 6401 7179 7955 8731 9504 0994 i 17S2 2568 . 3353 4136 4919 5699 6479 7256 8033 1073 .560, ?^^ w?A 0971 1$M 1741 1048 1125 1813 18^5 t^vi 2433 25<.'9 25S6 2663 1860 2647 3431 4215 4997 5777 , 6 S556 ' 7 7384 ; 8 8110 ; 9 8 ' 9 : ' SS08 8885 560 9582 9659 1 .»45 0:23 02i>0 0277 0354 0431 j*^,"*;*? ;:« y.-i -4.. 4 \ir'r :«^ :*.*.; \s'.'j I r:-4* 1115 ir-i r*r* « !:•:•? i>i^ *r-.; ^;4T 3:23 3>» 3277 3353 8430 3506 7 4^<! }.-r.> i'M 3.*: 3 * 4M? 44i5 4J- : 457* 1202 1972 2740 3^!^ 3v^ 4<:42 4119 4195 4272 46J4 473*) 4><>7 4883 4960 5036 9 "Hi £:i* 5it5 5341 5417 5494 5570 5646 5722 6799 9 V : .- 3 4 5 6 7 8 9 570 :.*:: :Vi; t->:7 fi-.'VS 6:j;i 6256 6.^2 6403 6484 6560 570 ^ A e-^- ■>::2 ?"ji 6>o4 6:='40 7«>16 7092 7163 7244 7820 o r;-^. 7472 754? 7624 77C-1 7775 7S51 7927 8008 6079 3 s::<- *i:?^j S.^>o 5&S2 84^8 S533 wk»9 8685 8761 8836 4 s^:i s^ss s--a ^•139 y2l4 92y<i 9366 9441 9517 9592 5 ««* t743 9?1> 9S94 9k'70 0799 0121 0S75 0196 0950 0272 1025 0347 1101 ! 6: 6'>422 •>49S 0573 IV849 0724 7 ri:«5 1251 132^ 1402 1477 1552 1627 1702 1778 1858 : 8 i**j* 2.i«>3 2»»75 2153 222S 23'.'3 2378 2453 2529 2604 9 2679 2754 2529 29i'4 2978 3053 3128 3203 3278 3853 1 2 3 4 5 6 7 8 9 580 8428 3503 3578 3653 8727 3802 3877 8952 4027 4101 5S0 ' 1 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 1 . 2 4923 4998 5*172 5147 5221 5296 5370 5445 5520 5594 2 • 8 5669 5743 5S13 5892 5966 6041 6115 6190 6264 6838 8 4 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 4 , 5 7156 7230 73rt4 7379 7453 7527 7601 7675 7749 7828 5 i 6 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 6 7 8638 8712 8786 8860 8934 9008 9082 9156 9230 9803 ( 8 9377 9451 9525 9699 9673 9746 9820 9894 9968 0042 r 8 9 770116 0189 0263 0336 0410 0484 0557 0631 0705 0778 9 ; 1 2 3 4 5 6 7 8 9 690 0852 0926 0999 1073 1146 1220 1298 1867 1440 1514 590 1 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 1 •2 2 2322 2395 2468 2542 2615 2688 2762 2885 2908 8981 8 8055 3128 3201 8274 3348 8421 3494 8567 3640 3718 S 4; 8786, 3860 8933 4006 4079 4153 4225 4298 4371 4444 4 6. 46171 4590 4663 4786 4809 4882 4955 5028 5100 5178 5 6 5246, 5819 5892 5465 5538 5610 5683 5756 5829 5902 6 5 5974 1 6047 6120 6193 6265 6838 6411 6488 6556 66S9 7, 6701 6774 6846 6919 6992 7064 7137 7209 7888 7354 8 ol 7427 7499 7572 i 7644 7717 1 7789 7862 7984 8006 1 807» 9 11 Hi. 6000 L. 778161] TABLE III. [No. 6499 L. 812847 N. 1 2 8 4 5 6 7 8 9 N. 600 600 778161 8224 8296 8868 8441 8513 8686 8668 8780 8802 1 8874 8947 9019 9091 9163 9286 9308 9380 9452 9624 1 2 9096 9669 9741 9818 9885 9957 0029 0749 0101 0821 0173 0893 8 780817 0889 0461 0538 0605 0677 9246 0965 2 8 4 1087 1109 1181 1253 1324 1896 1468 1640 1612^ 1684 4 5 1755 1827 1899 1971 2042 2114 2186 2268 2329 2401 5 6 7 2473 2544 2616 2688 2759 2831 2902 2974 3046 8117 6 8189 8260 8882 8408 3475 3546 3618 8689 3761 8832 7 8 8904 8975 . 4046 4118 4189 4261 4382 4408 4475 4646 8 9 4617 4689 1 4760 4831 4902 4974 6046 6116 6187 6269 9 6 1,2 3 4 5 6 7 8 9 610 5880 5401 ! 5472 5543 5615 5686 6767 5828 5899 5970 610 1 60411 6112 6188 6254 6325 6396 6467 6538 6609 6680 1 2 67511 6822 ; 6893 6964 7035 7106 7177 7248 7319 7390 2 8 7460, 7581 7602 7673 7744 7815 7885 7966 8027 8098 8 4 8168' 8289 ! 8310 8381 8451 8522 8693 8663 8734 8804 4 6 8875 8946 9016 9087 9167 9228 9299 9369 9440 9610 5 6 9581 9651 9722 9792 9863 9933 0004 0707 0074 0778 0144 0848 7 790285 0856 0426 0496 0567 0637 0216 0918 6 7 8 0988 1059 1129 1199 1269 1340 1410 1480 1560 1620 8 9 1691 1761 1881 1901 1971 2041 2111 2181 2262 2322 9 1 2 3 4 5 6 7 8 9 620 2892 2462 2532 2602 2672 2742 2812 2882 2962 8022 620 1 ' 8092 8168 8231 3301 3371 3441 3611 8581 8661 3721 1 2 8790 8860 3980 4000 4070 4139 4209 4279 4349 4418 2 8 4488 4558 ! 4627 4697 4767 4836 4906 4976 6045 6115 8 4 5185 5254 5324 5398 5468 5632 6602 6672 6741 5811 4 5 5880 5949 6019 6088 6158 6227 6297 6366 6486 6506 5 6 6574 6644 6713 6782 6852 6921 6990 7060 7129 7196 6 7 7268 7887 7406 7475 7545 7614 7683 7762 7821 7890 7 8 7960 8029 8098 8167 8286 8805 8874 8443 8618 8582 8 9 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 9 1 2 3 4 5 6 7 8 9 10 9841 9409 9478 9547 9616 9686 9764 9823 9892 9961 630 1 I 800029 0098 0167 0236 0306 0373 0442 0611 0680 0646 2 0717 0786 0854 0923 0998 1061 1129 1198 1266 1885 2 \ , 1404. 1472 1541 1609 1678 1747 1815 1884 1952 2021 3 2089 2158 2226 2295 2363 2432 2600 2668 2637 2706 4 •774 2842 2910 2979 3047 3116 3184 3262 3321 3389 5 8457 8525 8594 8662 3730 3798 8b67 3935 4003 4071 6 4189 4208 4276 4844 4412 4480 4548 4616 4685 4753 7 4821 4869 4967 5025 5093 5161 5229 6297 5365 5438 8 5501 5569 5637 5705 6778 6841 5908 5976 6044 6112 9 1 2 8 4 5 6 7 8 9 6180 6248 6816 6384 6451 6619 6587 6655 6728 6790 640 6858 6926 6994 7061 7129 7197 7264 7832 7400 7467 1 7585 7603 7670 7788 7806 7878 7941 8008 8076 8148 2 8211 8279 8346 8414 8481 8649 8616 8684 8761 8818 3 8886 8958 9021 9088 9156 9223 9290 9368 9425 9492 4 9560 9627 9694 9762 9829 9896 9964 0031 0703 0098 0770 >2S8 0800 0367 0434 0501 0669 0636 0165 0837 5 6 »904 0971 1039 1106 1178 1240 1807 1374 1441 1508 7 575 1642 1709 1776 1848 1910 1977 2044 2111 2178 8 M5 2312 2379 2445 2512 2579 2646 2718 2780 2847 9 1-2 1359; 1422 1985 2047 2609 2672 3283 3295 8855 3918 4477 4589 0232 0859 1485 2110 , 2734 I 8857 8980 4601 13 No. N. 700 7000 L. 845098] TABLE III. [No. 7499 L, 876003 1 2 3 4 5 6 7 8 9 N. 845098 5160 ^ 5222 5284 5346 5408 5470 5532 5594 5656 700 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 1 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 2 3 6955 7017 7079 7141 7202 7264 7826 7388 7449 7511 8 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 4 6 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 5 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 6 7 8 9419 9481 9542 9604 9665 9726 9788 0401 9849 9911 9972 7 8 850083 0095 0156 0217 0279 0340 0462 0524 0585 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 9 1 2 3 4 6 6 7 8 9 710 1258 1320 1381 1442 1508 1564 1625 1686 1747 1809 710 1 1870 1981 1992 2058 2114 2175 2236 2297 2358 2419 1 2 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 2 8 8090 8150 8211 3272 8333 3394 3455 3516 8577 3637 8 4 8698 3759 3820 3881 3941 4002 4063 4124 4185 4245 4 6 4306 4867 4428 4488 4549 4610 4670 4731 4792 4852 5 6 4913 4974 5034 5095 5156 5216 5277 5337 5898 5459 5 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 7 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 8 9 6729 6789 6850 6910 6970 7081 7091 7152 7212 7272 9 1 1 2 8 4 5 6 7 8 9 720 7382 7893 7458 7518 7574 7684 7694 7755 7815 7875 720 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 1 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 2 8 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 8 4 9739 9799 9859 9918 9978 0038 0637 0098 0697 0158 0757 0218 0817 0278 0877 4 5 5 860838 0398 0458 0518 0578 6 0987 0996 1056 1116 1176 1236 1295 1355 1415 1475 6 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 7 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 8 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3268 9 1 2 3 4 5 6 7 8 9 780 8828 8382 3442 3501 3561 3620 3680 3739 3799 3858 780 1 8917 8977 4036 4096 4155 4214 4274 4833 4892 4452 1 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 2 8 6104 5168 5222 5282 5341 5400 5459 5519 6578 6687 3 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 4 5 6287 6846 6405 6465 6524 6583 6642 6701 6760 6819 6: 6 6878 6937 6996 7055 7114 7173 7282 7291 7350 7409 6 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 7 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 8 9 8644 8708 8762 8821 8879 8938 8997 9056 9114 9173 9 1 2 3 4 5 6 7 8 9 740 9282 9290 9349 9408 9466 9525 9584 9642 9701 9760 740 1 9818 9877 9935 9994 0053 0638 0111 0696 0170 0755 0228 0813 0287 0872 0345 0930 1 2 2 870404 0462 0521 0579 8 0989 1047 1106 1164 1223 1281 1339 1898 1456 1516 8 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 4 6 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 5 6 2739 2797 2855 2913 2972 3030 3088 8146 3204 3262 6 7 8821 8879 8487 8495 3553 3611 3669 3727 8785 8844 7 8 8902 8960 4018 4076 4134 4192 4250 4308 4866 4424 8 9 4482 4540 4598 4656 4714 4772 4880 4888 4945 5008 9| 14 Ve. 7M0 L. 875061] TABLB III. [Ifo. 7000 L. 0O8O86| 750 1 2 % 4 5 6 7 8 875061 5119 5177 1 6285 5298 5851 5400 6466 5524 5582 ,750 1 5640 5698 5756 581S 5871 6980 5087 6045 6108 6160 1 1 2 6218 6276 6338 1 6891 6449 6507 6564 6622 6680 6787 1 2 8 6795 6853 6910 : 6968 7026 7088 7141 7100 7266 7814 ' 3 4 7871 7429 7487 7544 ■ 7602 7659 7717 7774 7882 7889 , 4i 5 7947 8004 8062 8110 j 8177 8284 8202 8840 8407 8464 5' 6 8522 8579 8637 < 8694 > 8752 8809 8866 8024 8981 9039 1 6 7 9096 9158 0211 0268 1 0825 9888 0440 0407 9565 9612 7 8 9669 9726 0784 9841 9898 9956 1 0018 0070 0127 ' 0185 8 9 880242 0299 0856 0418 0471 0528 0585 0642 0600 J 0756 9 1 2 8 4 6 6 7 8 , • ' 760 0814 0871 0028 0085 1042 1099 1156 1218 1271 , 1828 760 1 1885 1442 1499 1556 1613 1670 1727 1784 1841 * 1898 1 2 1955 2012 2069 2126 2183 2240 2297 2854 2411 1 2468 ' 2 8 2525 2581 2<J38 ; 2695 | 2752 2809 2866 2028 2980 80S7 3 4 8093 8150 8207 8264 ; 8321 3377 8484 8491 8548 8605 4 5 8661 3718 8775 8882 i 8888 8945 4002 4059 4115 41T2 > 5 6 4229 4285 4842 4899 ; 4455 4512 4569 4625 4682 47S9 6 7 4795 4852 4909 ! 4965 i 5022 5078 5135 5192 5248 1 5805 7 8 5861 5418 5474 i 5581 5587 5644 5700 5757 5818 5870 8 5926 5983 6039 6096 6152 6209 6265 6821 6878 6434 9 1 2 1 8 4 5 6 7 8 t 770 6491 6547 6604 6660 6716 6778 6829 6885 6942 6908 770 1 7064 7111 7167 1 7223 7280 7336 7392 7449 7505 7561 1 2 7617 7674 7730 '■ 7786 7842 7898 7955 8011 8067 8123 2 8 8179 8236 8292 ' 8348 8404 8460 8516 8578 8629 8685 ' 8 4 8741 8797 8858 ; 8909 8965 9021 9077 9184 9190 9246 i 5 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 . 5 9862 9918 1 9974 0086 0645 0141 0700 0197 0756 0253 0812 0309 ; 0365 « 0868 I 0924 7 7 890421 0477 0533 0589 8 0980 1035 1091 1147 1208 1259 1314 1370 1426 1482 8 1537 1593 1649 1705 1760 1816 1872 1928 1083 2039 r 1 2 i 8 4 5 6 7 8 1 ' 780 2095 2150 2206 i 2262 2317 2873 2429 2484 2540 2505 T80 1 2651 2707 2762 i 2818 2873 2929 2985 8040 8006 8151 1, 2 3207 8262 3818 ': 8373 3429 8484 8540 8505 3651 8706. J 8 8762 3817 3878 8928 8984 4089 4094 4150 4205 4861 S 4 4316 4371 4427 ; 4482 4538 4593 4648 4704 4759 4814 < 5 4870; 4925 4980 5086 5091 5146 5201 5257 5812 5867- \ 6 5423 5478 5533 i 5588 5644 5699 5754 5800 5864 5920, • 7 5975 6080 6085 : 6140 6195 6251 6806 6861 6416 6471 ' 8 6526 6581 6636 6692 6747 6802 6857 6912 6067 7022 8 7077 7182 7187 7242 7297 7852 7407 7462 7517 7572 », 1 2 3 4 5 6 7 8 • 700 7627 7682 7787 ; 7792 7847 7902 7957 8012 8067 8122 ITW! 8670 1 1 8176 8281 8286 8341 8396 8451 8506 8561 8615 e 8725 8780 8885 8890 8944 8999 9054 9109 9164 0218 8 0278 0828 9863 9487 9492 0547 9602 0656 0711 9766 4 0821 9875 9980 9985 0089 0586 0004 0640 0149 0695 0808 0740 0258 0804 0312 0850 5 000867 0422 0476 0581 a 0918 0068 1022 1077 1181 1186 1240 1805 1840 1404 r 1458 1518 1567 1682 1676 1781 1785 1840 1804 1048 8 2008 2057 2112 8166 8221 2275 2829 2884 2488 2198 2647 2601 2655 8710 2764 8818 8878 2027 2081 8686 __\ 15 N«. 8000 L. 908090] TABLB III. [ir«. 8499 L. 9298n! N. 1 2 8 4 6 6 r 8 9 H. 800 908090 8144 8199 8258 8307 3861 3416 8470 8524 8578 800 1 8888 8687 8741 8795 3849 3904 3958 4012 4066 4120 1 2 4174 4229 4288 4887 4391 4445 4499 4663 4607 4661 2 8 4716 4770 4824 4678 4932 4986 5040 5094 5148 6202 4 5256 6810 5864 5418 5472 5526 5580 5684 5688 5742 6 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 6 6885 .6889 6448 6497 6551 6604 6658 6712 6766 6820 7 6874 6927 6981 7086 7089 7148 7196 7250 7804 7868 8 7411 7465 7519 7578 7626 7680 7784 7787 7841 7895 8 9 7949 8002 8056 8109 8163 8217 8270 8824 8878 8431 9 1 2 8 4 5 6 7 8 9 .810 8485 8589 8592 8646 8699 8768 8807 8860 8914 8967 810 1 9021 9074 9128 9181 9285 9289 9842 9896 9449 9603 ' ll 2 8 9556 9609 9663 9716 9770 9828 0858 98n 9980 OOfil 0037 0571 2 8 910091 0144 0197 0251 0804 0411 0464 0618 4 0624 0678 0781 0784 0888 0891 0944 0998 1051 1104 4 ' 6 1158 1211 1264 1817 1871 1424 1477 1630 1584 1687 5 6 1690 1743 1797 1850 1903 1956 2009 2063 • 2116 2169 6 7 2222 2276 2828 2881 2485 2488 2541 2694 2647 2700 7 8 2758 2806 2859 2918 2966 8019 8072 8126 3178 8281 8 9 8284 8887 8890 8443 8496 8549 8602 8666 8708 8761 9 1 2 8 4 5 6 7 8 9 820 8814 8867 8920 3973 4026 4079 4132 4184 4287 4290 820 1 4848 4896 4449 4502 4555 4608 4660 4713 4766 4819 1 2 4872 4925 4977 5030 5083 5186 5189 6241 6294 5347 2 S 5400 5458 5505 5558 5611 5664 6716 5769 5822 5875 3 4 5927 5980 6083 6085 6188 6191 6248 6296 6349 6401 4 6 6454, 6507 6559 6612 6664 6717 6770 6822 6875 6927 5 6 6980 7088 7085 7138 7190 7248 7295 7348 7400 7 last ({ 7 7506 7558 7611 7663 7716 7768 7820 7873 7926 7978 7 a 8080 8088 8185 8188 8240 8293 8846 8397 8450 ' 8602 8 9 8555 8607 8659 8712 8764 8816 8869 8921 8978 9026 9 1 2 8 4 5 6 7 A g 830 9078 9180 9188 9285 9287 9340 9392 9444 9496 9549 880 1 2 ^601 9658 9706 9758 9810 9862 9914 9967 1 2 0019 0541 0071 0698 920128 0176 0228 0280 0332 0384 0436 0489 S 0645; 0697 0749 0801 0853 0906 0968 1010 1062 1114 8 4 11661 1218 1270 1322 1374 1426 1478 1630 1582 1634 4 6 1686 1788 1790 1842 1894 1946 1998 2060 2102 2154 5 6 2206. 2258 2810 2362 2414 2466 2618 2670 2622 2674 5 7 27251 2777 2829 2881 2983 2985 8087 8089 8140 8192 7 8 8 3244: 8296 8848 8399 8451 8508 8665 8607 8668 8710 9 8762, 8814 8865 8917 3969 4021 4072 4124 4176 4228 9 1 2 8 4 5 6 7 8 9 4744 840 4279 4881 4888 4434 4486 4588 4689 4641 4698 840 1 4796 4848 4899 4951 5003 5054 5106 6157 6209 5261 1 2 2 5812 5864 5415 5467 5518 5570 5621 5678 5725 5776 8 5%2» 5879 5981 5982 6084 6085 6187 6188 6289 6291 8 4 5 6 7 8 9 4 6842 6894 6445 6497 6548 6600 6651 6702 ; 6764 * 6805 6 6857 6908 6959 7011 7062 7114 7166 7216 i 7268 7819 7832 8 7870 7422 7478 7524 7576 7627 7678 7780 1 7781 7 7888 7985 7986 8087 8088 8140 8191 8242 8298 8845 8 8896 8447 8498 8549 8601 8652 8708 8754 8805 8857 9 8908 8959 9010 9061 9112 9163 9S15 9266 19817 9868 16 ,No. S5iiO L. 929419] TABLE III. [No. 8999 L. 954194 1240 , 1289 ?i 1726 ' 1776 { 2211 . 2259 5 2696 I 2744 < 3180 I 3228 : , 866S i 8711 ^i 4146 I 4194 I ' 17 5«. 9000 L. W42i«\ TABLB XXL Pf 0. 9499 L. 977678 K. 1 2 8 4 5 6 7 8 9 N. 990 984248 4291 4339 4387 4436 4484 4582 4580 4628 4677 900 1 4726 4778 4821 4869 4918 4966 5014 5062 5110 5168 1 2 8207 5288 8308 5352 5399 5447 5495 5543 5692 5640 2 3 8888 5736 5784 5882 5880 5928 5976 6024 6072 6120 3 4 8188 6216 6265 6313 6861 6409 6457 6505 6653 6601 4 5 8649 6697 6745 6798 6840 6888 6936 6984 7032 7080 5 6 7128 7176 7224 7272 7320 7868 7416 7464 7612 7559 6 7 7807 7688 7703 7751 7799 7847 7894 7942 7990 8038 7 8 8086 8184 8181 8229 8277 8825 8373 8421 8468 8516 Sj 9 8864 8612 8659 8707 8755 8803 8850 8898 8946 8994 9 1 2 3 4 6 6 7 8 9 910 9041 9089 9137 9185 9282 9280 9328 9375 9423 9471 910 1 9818 9586 9614 9861 9709 9757 9804 9852 9900 9947 1 ^i •fifl/k wvo 0042 0518 0090 0566 0138 0613 0185 0661 •233 0709 0281 0766 0328 0804 6376 0851 0428 0899 2 3 8 980471 4 0946 0994 1041 1089 1136 1184 1281 1279 1326 1874 4 6 1421 1489 1816 1863 1611 1658 1706 1753 1801 1848 5 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 6 7 2889 2417 2464 2511 2559 2606 2663 2701 2748 2796 7 8 2848 2890 2987 2988 3032 8079 8126 3174 8221 3268 8 9 8816 3863 3410 3457 8504 8552 8599 3646 3693 3741 9 1 2 3 4 5 6 7 8 9 ftIO 8788 3838 3882 3929 3977 4024 4071 4118 4165 4212 920 1 4260 4807 4854 4401 4448 4495 4542 4690 4637 4684 1 2 4781 4778 4825 4872 4919 4966 5018 5060 5108 6155 2 8 8202 5249 5298 5343 5890 5487 6484 5631 6678 5625 8 4 8872 5719 5766 5813 5860 6907 5954 6001 6048 6096 4 5 8142 8189 6286 6283 6829 6376 6428 6470 6617 6664 6 8 6611 6658 6705 6752 6799 6845 6892 6939 6986 7083 6 7 7080 7127 7178 7220 7267 7314 7361 7408 7464 7601 7 8 7848 7595 7642 7688 7736 7782 7829 7876 7922 7969 8 9 8016 8062 8109 8156 8203 8249 8296 8348 8389 8486 9 1 2 8 4 6 6 7 8 9 980 8483 8580 8576 8628 8670 8716 8763 8810 8866 8903 930 1 8980 8998 9048 9090 9186 9183 9229 9276 9823 9869 1 2 9416 9463 9509 9556 9602 9649 9695 9742 9789 9836 2 8 9882 9928 9975 0021 0486 0068 0588 0114 0579 0161 0626 0207 0672 0254 0719 0800 0765 3 4 4 970847 0398 0440 6 0812 0858 0904 0951 0097 1044 1090 1187 1188 1229 5 6 12T6 1822 1869 1415 1461 1608 1564 1601 1647 1698 6 7 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 7 ( 2203 2249 2295 2842 2888 2434 2481 2527 2678 2619 8 fl 2666 2712 2758 2804 2851 2897 2943 2989 8085 8082 9 1 2 8 4 5 6 7 8 9 I4C 8128 3174 8220 3266 3813 8869 3405 3451 8497 8543 940 1 8880 3636 8682 3728 3774 8820 8866 3913 8969 4005 1 3 4081 4097 4148 4189 4286 4281 4327 4374 4420 4466 2 2 4812 4858 4604 4650 4696 4742 4788 4834 4880 4926 3 4 4972 8018 8064 5110 5156 5202 5248 5294 5840 5886 4 e 8482 8478 8524 5570 5616 5662 5707 5753 5799 5846 6 fl 8891 8987 8988 6029 6075 6121 6167 6212 6258 6304 6 7 8880 6396 6442 6488 6538 6579 6625 6671 6717 6763 7 8 MOO 8854 6900 6946 6992 7037 7083 7129 7175 7920 8 9 7886 7812 7858 7408 7449 7495 7641 7586 7632 7678 9 18 No. 9600 L. 977724] TABLE III. [No. 9999 L. 999987! N. 1 2 8 4 6 6 7 8 9 N. 050 977724 7769 7816 7861 7906 7962 7998 8043 8089 8185 950 1 8181 8226 8272 8817 8863 8409 8464 8600 8546 8591 1 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 S 3 9093 9138 9184 9230 9276 9321 9366 9412 9457 9503 8 4 5 9548 9594 9639 9686 1 9730 9776 9821 9867 9912 9958 4 980003 0049 0094 0140 ' 0186 0281 0276 0822 0867 0412 6' 6 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 6, 7 0912 0957 1008 1048 1093 1189 1184 1229 1275 1820 7l 8 1366 1411 1456 1501 1647 1692 1637 1688 1728 1778 1 8 9 1819 1864 1909 1954 1 2000 1 2045 2090 2185 2181 2226 9 1 1 2 8 4 6 6 7 8 • 1 960 22n 2816 2362 2407 2462 2497 2643 2588 2688 2678 |960| 8180 ll 1 2728 2769 2814 2859 2904 2949 2994 8040 ■ 8085 2 8175 3220 8266 8310 8856 8401 8446 8491 ' 3586 8581 1 Si 8 8626 3671 8716 3762 i 8807 8852 8897 8942 , 8987 4082 S- 4 4077 4122 4167 4212 . 4257 4802 4847 4892 1 4487 4482 4i 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4982 6; 6 4977 6022 6067 6112 6167 6202 6247 6292 5837 5882 6' 7 5426 6471 6516 6561 6606 6651 6696 6741 5786 5880 1 7i 8 6876 5920 6965 6010 6056 6100 6144 6189 1 6284 6279 1 Si 9 6324 6369 6418 6458 6593 6548 6598 6687 6682 6727 9 1 2 8 4 6 6 7 8 » 970 6772 6817 6861 6906 1 6951 6996 7040 7085 7180 7176 970 1 7219 7264 7309 7863 ! 7398 7448 7488 7682 7577 7622 1 2 7666 7711 7766 7800 7846 7890 7984 7979 8024 8068 i 8 8113 8157 8202 8247 8291 8386 8881 8425 , 8470 8514 i 4 8559 8604 8648 8698 8737 8782 8826 8871 1 8916 8960 4 6 9005 9049 9094 9138 ' 9188 9227 9272 9816 9361 9405 5 6 9450 9494 9639 9688 9628 9672 9717 9761 9806 9850 « 7 9895 9939 9983 0028 0072 0472 1 0616 0117 0561 0161 0605 0206 1 0250 0650 0694 0294 r 0788 8 8 990339 0388 0428 9 0783 0827 0871 0916 0960 1 1004 1049 1098 1187 1182 9 1 2 8 4 6 6 7 8 9 980 1226 1270 1316 1869 1408 1448 1492 1586 1580 1625 980 1 1669 1718 1768 1802 1846 1890 1935 1979 2028 2067 1 2 2111 2166 2200 2244 2288 2888 2877 2421 2465 2509 i 8 2564 2598 2642 2686 2730 2774 2819 2868 2907 2961 8 4 2995 3039 3088 3127 8172 8216 8260 8804 8848 8892 j 4 5 8436 8480 8524 8668 8618 8667 8701 8745 8789 88331 6 6 8877 8921 8966 4009 4068 4097 4141 4185 4229 4878 1 « 7 4817 4361 4406 4449 4498 4687 4681 4625 4669 4718; 7 8 4767 4801 4846 4889 1 4938 4977 6021 5065 5108 5158 1 8 9 6196 6240 6284 6328 ! 6372 6416 6460 5504 5547 5591 9 1 2 1 8 1 4 6 6 7 8 9 990 6685 6679 6728 6767 ! 6811 6864 6898 5942 5986 6080 990 1 6074 6117 6161 6206 6249 6298 6887 6880 6424 6468 1 2 6612 6566 6699 6648 6687 6781 6774 6818 6862 6906 S 8 6949 6998 7037 7080 7124 7168 7212 7255 7299 7848 8 4 7886 7480 7474 7617 7661 7606 7648 7692 7786 7779 4 6 7823 7867 7910 7964 7998 8041 8086 8129 8172 8216 f 6 8269 8808 8847 8890 8484 8477 8621 8564 8608 8652 6 7 8696 8789 8782 8826 8869 8918 8966 9000 9048 9087 T 8 9180 9174 9218 9261 9806 9848 9892 9485 9479 9528 8 9 9666 ! 9609 9662 9696 9789 9788 9826 9870 9918 9957 9 10 Deo. TABLB lY.— Loo. Snris, no. Sine 86 87 88 89 40 41 48 086965 48 8.097188 44 107167 "' 116926 8 866816 ;--" 9 417968 lllll 10 463725 lll^ 11 605118 2S2? 12 542906 ^"^^^ 18 7.577668 ^J^JJ 14 609858 f3«*J 16 689816 lll^l 16 667846 ^Jj* 17 694178 «;5J 18 7189971 ll^ll 19 7.742477^ ^\^ 20 764754! llltL 21 28 825451 |2175 24 843984 K 25 7.861662 „„«» 26 878695. *?|?; 27 895085! -Jj^i 28 910879 JSoo 29 926119 *JJS ^ .!tr«o ^^^ 81 7.955082 ^ 82 968870 ^g 188 982288' *J*JS !84 995198 *1J2? 85 8.007787j IgSSJ 86 020021! ?2!?i S.081919 043501 054781 19802 18801 076500 ii?I* 076500 086965 ,^ I N "'"— ■ *- 45 J.XVV2V 46 126471 47 185810 48 144958 50 162681 171280 j:^ 49 8.158907 51 1 # xxov 179718 ^I^SS » 187985 J^XX 54 196102 \f^ K 8.204070 laJT; 56 211895 J22JJ 57 219581 }|!iS 58 227184 {^4 19 284557 igS «J 2 41855 "^^ 17441 17081 16689 16265 15908 15566 15288 14924 14622 14888 COMC. Infinite. 13.586274 285244 059158 12.984214 837304 758128 12.691176 633184 582082 536275 494882 457094 12.422382 890147 860184 332155 805827 281008 12.257523 235246 214057 193854 174549 156066 12.188338 121805 104915 089121 078881 059158 12.044918 081180 017767 004802 11.992218 979979 11.968081 956499 945219 984224 928500 918035 11.902817 892883 888074 878529 864190 855047 11.846098 887819 828720 820287 812015 808898 11.795980 788105 780419 772866 765448 758145 Tang. D. 0.000000 6.463726 764756 940847 7.066786 162696 241878 7.808825 866817 417970 463727 505120 542909 7.577672 609857 639820 667849 694179 719003 7.742484 764761 785951 806155 825460 843944 7.861674 878708 895099 910894 926134 940858 7.955100 968889 982253 995219 8.007809 020045 8.031945 048527 054809 065806 076581 086997 8.097217 107202 116963 126510 185851 144996 8.153952 162727 171328 179763 188036 196156 8.204126 211953 219641 227195 284621 241921 601717 293485 208281 161517 131969 111578 96653 85254 76263 68988 62981 57938 58642 49989 46715 43882 41373 39136 37128 85316 33673 32176, 308071 29549 28890 27318 Cotang. Secant Ootang. 254011 245401 28785 22982 22276 21610 20983 20392 Infinite. 13.636274 285244 069153 12.934214 837304 768122 12.691176 633188 582030 536278 494880 467091 12.422328 390143 860180 832161 805821 280997 12.257516 235239 214049 193845 174640 156066 12.138326 121292 104901 089106 073866 069142 12.044900 081111 017747 004781 11.992191 979966 19888' 198051 ^^•^^^^^'^ J?r!!xl 966473 18808 I8327I 17875 17444 17084 945191 934194 923469 913003 16642; ll-»02783 16268, 15912 15568 16241 149271 14625 148861 14067| 13790, 18532, 182841 13044 12814J 12590'| 12876 12168 D. 892798 883037 873490 864149 865004 11.846048 887273 828672 820287 811964 803844 11.795874 788047 780359 772805 766879 758079 Secant D.l Cosine 10.000000 1 10.000001 1 1 2 2 8 10.000003 4 4 6 5 6 10.000007 7 8 9 10 11 10.000011 12 13 14 15 17 10.000018 19 20 21 28 24 10.000025 27 28 29 81 82 10.000084 36 87 89 41 42 10.000044 46 48 50 52 54 10.000056 58 60 62 64 66 Tang. CoMC. 10.000000 9.999999 9.999999 9 9 8 8 7 9.999997 6 6 5 5 4 9.999993 3 2 1 89 9.999989 88 87 86 85 83 9.999982 81 80 79 77 76 9.999975 78 72 71 69 68 9.999966 64 68 61 59 58 9.999956 54 52 50 48 46 9.999944 42 40 88 86 84 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 86 85 84 38 82 81 30 29 28 27 26 25 24 28 22 21 20 19 18 17 16 15 14 18 12 11 10 9 8 7 6 5 4 8 2 1 Sine 89 Dto. 20 1 Deg. TABLE lY.— Loo. 8iffB, im Sin* Coflec. Tang. Cotang. ,8.241855 1 I 41*033 2 , 56«>y4 63042 69SS1 7t>614 83243 7 . 8.289773 8 96207 9 8.302546 10 . 08794 U. 14954 12 , 21027 18 8.327016 14 , 32924 15. 16 17. 18: 38753 44504 501S1 55783 19 8.361315 20. 21 22! 23 j 24 I 66777 72171 774in» 82762 87962 25 8.393101 26 98179 27.8.403199 28 29 80 08161 13068 17919; 32 31 ; 8.422717| 274621 32156: 36800 41394' 45941: 34 I 35 I 36 t ,^..., 87 : 8.450440 88 i 54893 39! 401 42 I 59801 63665] 67985; 72263, 44 45 46 47 48 49 50 51 52 53 54 55 56 ,67 i58 59 60 80693 84848 88963 93040 97078 8.501080 05045 08974 12867 16726 20551 8.524843 28102 81828 85528 89186 42819 11963 1176S 115S0 1139S 11221 11050 10883 10723 10565 10413 H.»266 10122 9982 9S47 9714 9586 9460 9338 9219 9103 8990 8880 ; 8772 i 8667 j 8564 8464 8366 8271 81771 8086, 7996; 7909 ; 7823 ! 7740 7657 7577 7499 7422 7846 7273 7200 7129 7060 6991 6924 6859 6794 6781 6669 6608 6548 6489 6482 6375 6319 6264 6211 6158 6106 6055 11.758145 8.241921 50967 49102 43906 86958 30119 23386 16757 56165 63115 69956 76691 83323 11.710227 03793 11.697454 91206 85046 78973 11.672984 67076 61247 55496 49819 44217 11.638685 33228 27829 22501 17238 12033 11.606899 01821 11.596801 91839 86932 82081, 11.577283' 72538 67844 63200 58606^ 54059| 11.549560, 4510' 40699 86335 82015 2773 11.523502 1930! 15152 11037 06960 02922 11.498920 94955 91026 87133 83274 79449 11.47565' 71898 68172 64477 60814 57181 8.289856 96292 8.302634' 08884 15046 21122^ 8.827114' 33025 38856 44610. 50289 55895. 8.861430' 66895- 722921 77622' 82889! 88092; 8.893234- 98315; 8.4033381 08304! 13213J 18068{ 8.422869! 27618! 82315 86962 41560 :: 46110 8.450613 55070 59481! 638491 68172 72454 8.476693* 80892 85050j 89170 03250 97293 8.501208 05267 09200 13098 16961 20790 8.524586 88349 82080 85779 89447 43084 11967 11772 11584' 11402 11225' 11054' 10887 ■ 10726 10570 10418 10270 10126' 9987- 9851- 9719' 9590 9465 9343! 9224 9108 8995 8885 8777' 8672' 8570 8470, 8371' 8276 ' 8182! 8091 1 8002' 7914 7828 7745 7663 7583 r505 Seomt ID.' Counc I 7428 7352 7279 7206 7135 7066 6998 6931 6865 6801 6738 6676 6615 6555 6496 6439 6882 6826 6272 6218 6165 6118 6062 11.758079 5089d[ 43885| 86885, 80044 38309 166771 11.710144; 08708 I 11.697366| 91116 84954 78878- 11.672886 66975 61144 65390 49711 44105 11.688570 83105 27708 82878 17111 11908 11.606766 01685; 11.596662; 91696 867871 81982 11.577131! 72382; 67685 63038| 58440 53890 11.549387 44930| 40519' 86151! 818281 27546| 11.523807; 19108' 14950| 10830| 06750| 02707 10.000066 68- 71 i 78! 75! 80' 10.000082* 85 87? 90! 9s; 95; 10.000098 10.000101; c ^2 5 10.000115! , 8l'^ 24 271 80 10.000133 Wi 89 42 49j 10.000152 66 69i 62 66| 69 5 5 f! ilj ;i5; ,!«' 6 6 6 6 6 6 11.498702! 94738 90800 86902 88089 79210 11.475414 71651 67920 64221 60558 56916 10.000178; « 77|« 80| 84' 88 91 10.000195 99 10.000208 07 10 10.000218' 22; 26 81 85 89 10.000843 47 88 68 80 88 9.999984 60. 82 591 29 58 27 57' 85 56 82 55 ao;54 9.999918 > 58 , 15 52 18.51 10 50 07 49 05 48 9.999902; 47 9.999899 1 46 97i45 94 .44 91148 88:42 9.999885 1 41 82.40 79! 89 76 1 88 78.87: 70;8l! 9.999867 185 1 84.84,- 6ljl9< 68 8i| 54 '81 1 51 9.999848 44 41 tsiu' 84j» 81 24 1 9.999827 21: 90|»| 16|Mi i«'w| 9.999805,17, 01 IK 9.999797 j IS, »8!H. 90!IS 86-lS' 9.99978S;n 78:W 74I 9 69 65 61 9J90767 58 46 44 46 85 Cotiiie D. Secant Cotang. T^. 88 Did. 91 a Deg. D. ' j Sii»« I 4 670M llf? 5 60640 Sis 6 6»W9|^JJ^ 7-&M74S1; ^^. 10 11 Ifi GOMC. TABLE IV^Loo. Sins, xto. Cotang. Tang. D. Secant D. Codne 9.999786 60 81 I 69 26 68 22i67 17 '66 18 66 08 64 9.999704 ' 63 9.999699 62 94 61 89 60 86 49 80 48 9.999676 47 70 46 66,46 60 44 66 43 60,42 9.999646 41 40 40 36 89 29 88 24 87 19 86 9.999614 85 08.84 03 83 9.999697 82 92 81 86 80 9.999681 29 75,548 70 27 64 I 26 68 26 63|24 9.999647 I 28 41 I 22 86; 21 29 120 19 18 708W Sao 7*214 IS? 77666 ^l 80892 "** 12 84198, ^2 13 8.687469' ^^.^ 14 907211 "J; 16 98948) ?;;: 16 971621 ^l 17 8.600882 ^V 18 08489; l^ ao 09784i ^JJ5 12828 "J* M 26 26 27 28 29 90 84 88 6076 6041 6006 4972 18987 21962 26 &824066 SS854 J"* 86776 AQ9Q 80 89680 ^^ 82 46428 %Vi 48274 J?g "^<^ 4682 87 8.668476 .^« 88 62280 VSi 40 67689 1^ 41 70898 JJSJ 42 78080 J*S 48 &676761 ..q. ^ 78406 !!^ ^ ^IM' 4870 46 88666 !:;? 47 86272 :J:J 48 88868 Z^l 48 8.681488 .our 60 98998 VSit 61 96648 Jf;^ 62 99078 Jfjl 68 8.701689 ZVZ 64 04090 4!?? 66 8.706677 66 00049 67 68 68 "09049 JJIJ 11607 JJ5i 18962 AAKt 16888 J^ 18800 *^^ r©: 11.467181 68678 60006 46461 42946 89460 86001 11.482669 291641 267861 22484 191081 16807 11.412681 ' 092791 06062) 02848! 11.899668 96611, 11.898877 90266 871771 841091 810681 78088' 11.875086 72062: 66146 68224; 60820 11J67487 64672 61726 48898 46089 48298 11.840526 87770! 86082! 82811 29607 26920! 11J24249 21696i 189671 16886 18728 11187 11J08662 06002 08467 00927 11.298411 96910 11.298428 90961 884981 86048 88617 81200 8.648084 46691 60268 63817 67836 60828 64291! 8.667727: 711371 74620' 77877; 81208 84614 8.687796' 91061 94288 97492 8.600677: 03889! 8.606978 10094; 18189| 162621 19313' 22843; 8.626862: 28340, 81308> 84266 871841 40093 8US42982 46868 48704 61637 64862 67149 8.659928 62689 66483 68160, 70870 73668 8.676289 78900; 81644! 84172 86784 89381 J 8.691963 94629, 97081! 99617i 8.702189; 04646 8.707140! 09618; 12083 14634 16972 19896 Secant i Cotang. 6012 6962 6914 6866 6819 6773 6727 6682 6638 6695 6652 6510 6468 6427 6387 6847 6808 6270 6232 6194 6158 6121 6085 6060 6016 4981 4947 4913 4880 4848 4816 4784 4768 4722 4691 4661 4631 4602 4673 4644 4617 4488 4461 4484 4407 4380 4864 4828 4803 4277 4262 4228 4203 4179 4165 4182 4108 4086 4062 4040 D. 11.456916 58309 497